Habitat destruction, habitat restoration and eigenvector eigenvalue relations
|
|
- Anastasia Matthews
- 5 years ago
- Views:
Transcription
1 Mathematical Biosciences 181 (2003) Habitat destruction, habitat restoration and eigenvector eigenvalue relations Otso Ovaskainen * Department of Ecology and Systematics, Metapopulation Research Group, University of Helsinki, P.O. Box 65, Viikinkaari 1, FIN-00014, Helsinki, Finland Received 5 October 2001; received in revised form 10 April 2002; accepted 25 July 2002 Abstract According to metapopulation theory, the capacity of a habitat patch network to support the persistence of a species is measured by the metapopulation capacity of the patch network. Mathematically, metapopulation capacity is given by the leading eigenvalue k M of an appropriately constructed non-negative n n matrix M, where n is the number of habitat patches. Both habitat destruction (in the sense of destruction of entire patches) and habitat deterioration (in the sense of partial destruction of patches) lower the metapopulation capacity of the patch network. The effect of gradual habitat deterioration is given by the derivative of k M with respect to patch attributes and may be straightforwardly evaluated by sensitivity analysis. In contrast, destruction of entire patches leads to a rank modification of matrix M, the effect of which on k M may be derived from eigenvector eigenvalue relations. Eigenvector eigenvalue relations have previously been analyzed only for symmetric matrices, which restricts their use in biological applications. In this paper I generalize some of the previous results by deriving eigenvector eigenvalue relations for general non-symmetric matrices. In addition to the exact eigenvector eigenvalue relations, I also derive eigenvalue perturbation formulae for rank-one modifications. These results lead to simple and intuitive approximation formulae, which may be used e.g. to assess the contribution of particular habitat patches to the metapopulation capacity of the landscape. The mathematical results presented are not restricted to the metapopulation context, but they should find a number of useful applications in biology, engineering and other applied sciences, where the removal (or addition) of matrix rows and columns often corresponds in a natural manner to decreasing (or increasing) the degrees of freedom of the focal system. Ó 2003 Elsevier Science Inc. All rights reserved. Keywords: Eigenvalue eigenvector relation; Rank-one modification; Metapopulation dynamics; Habitat destruction; Habitat restoration; Patch value * Tel.: ; fax: address: otso.ovaskainen@helsinki.fi (O. Ovaskainen) /03/$ - see front matter Ó 2003 Elsevier Science Inc. All rights reserved. PII: S (02)
2 166 O. Ovaskainen / Mathematical Biosciences 181 (2003) Introduction One of the greatest challenges in the current development of biological theory is to account for various kinds of heterogeneities. Individuals are not identical, populations are not homogeneously mixed, and the world is not a regular lattice of identical sites. Often heterogeneity is of discrete nature and is most naturally described through a matrix. Consequently, matrix analysis has become an integral part of biological theory. A number of fundamental biological quantities have been found to appear as eigenvalues, examples including the growth rate of an age-structured population [1], the basic reproduction ratio [2] of an infectious disease, measures of community behaviour [3], the fitness of an individual [4] and the effective size of a subdivided population in population genetics [5]. Often one is interested not only in the absolute value of the quantity represented by the eigenvalue, but also in its sensitivity with respect to relevant model parameters. For example, in addition to knowing the current growth rate of an age-structured population, one would like to know the expected change in the growth rate that would follow from improving the survival of juveniles. As a consequence, the perturbation theory of eigenvalues is being increasingly used in the biological literature. In its most traditional form, eigenvalue perturbation theory is focused on the derivative of an eigenvalue, i.e., the change in an eigenvalue due to a small change in matrix elements. Such perturbation methods are most natural for considering the effect of small and gradual changes. However, in many biological applications, it is often relevant to consider large perturbations. As an example, I will study the following problem in this paper. Consider a fragmented landscape consisting of a discrete set of n habitat patches. According to metapopulation theory, the capacity of the patch network to support the persistence of a species is given by the metapopulation capacity k M of the fragmented landscape, defined as the leading eigenvalue of an appropriate Ôlandscape matrixõ M, which describes how the spatial structure of the habitat patch network (e.g. patch areas and interpatch distances) affects the colonization extinction dynamics of the species in the patch network [6]. A possible management scenario in such a landscape is the destruction of one (or some) of the habitat patches or, in the positive case, the restoration of a new habitat patch into a particular location. In order to evaluate the likely consequences of such management options, one should be able to estimate the change in the eigenvalue k M due to a rank modification of matrix M. For example, the destruction of a habitat patch corresponds to the deletion of a matrix row and a matrix column. Due to the singular nature of such a perturbation, approaches based on eigenvalue derivatives may not be applicable and eigenvalue eigenvector relations should be used instead. Eigenvector eigenvalue relations have been previously analyzed in the mathematical literature only for symmetric matrices, which restricts their applicability in many biological situations. The main contribution of this paper is given in Section 2, where I generalize some of the previous results by deriving eigenvector eigenvalue relations for non-symmetric matrices. As a corollary to the exact eigenvalue eigenvector relations, I derive approximation formulae for eigenvalue perturbations due to rank-one modifications. In particular, I show that if the dth row and the dth column of matrix A are deleted, the change in a simple eigenvalue k is given as k k ky d x d ; ð1þ
3 O. Ovaskainen / Mathematical Biosciences 181 (2003) where k is an eigenvalue of the modified matrix, and x and y are the right and left eigenvectors corresponding to the eigenvalue k, respectively. I also give an analogous formula for the change in a simple eigenvalue due to addition of a new row and a new column to the matrix. After presenting the mathematical theory in Section 2, I apply the eigenvector eigenvalue relations in Section 3 to study the effects of habitat destruction and habitat restoration in the metapopulation context. 2. Eigenvector eigenvalue relations Eigenvalue-eigenvector relations for rank-one modifications have previously been presented for symmetric matrices [7 9]. The principal result is that if A is a symmetric matrix and A is obtained by deleting the last row and the last column from A, it holds that Q n 1 ½x ðiþ n Š2 j¼1 ¼ ðk i k j Q Þ n j¼1;j6¼i ðk i k j Þ ; ð2þ where fk j g n j¼1 and fk j gn 1 j¼1 are the eigenvalues of A and A, respectively, and x ðiþ is the normalized eigenvector corresponding to a simple eigenvalue k i. I start by showing that a formula analogous to Eq. (2) holds for non-symmetric matrices as well. Although in most biological applications the matrix will be non-negative, and the eigenvalue of interest will be the leading eigenvalue, I state the result for simple eigenvalues of complex valued matrices, as the proof is not more difficult for the general case. I also give a formula analogous to Eq. (2), which holds for the addition of a new row and a new column to matrix A. Theorem 2.1. Let A 2 C nn, and let A 2 C ðn 1Þðn 1Þ be the matrix obtained by deleting the dth row and dth column from A, and let A þ be the matrix obtained by adding a new row and a new column to A, A þ ¼ A a C a H R a ; where the superscript H denotes conjugate transpose. Let fk i g n i¼1, fk i g n 1 i¼1 and fkþ i gnþ1 i¼1 be the eigenvalues of A, A and A þ, respectively. Let x ðiþ and y ðiþ be the right and left eigenvectors corresponding to a simple eigenvalue k i, normalized as y ðiþh x ðiþ ¼ 1. Then Q n 1 y ðiþ d xðiþ j¼1 d ¼ ðk i k j Q Þ n j¼1;j6¼i ðk i k j Þ ; ð3þ Q nþ1 ðy ðiþh a C Þða H R xðiþ j¼1 Þ¼ ðk i k þ j Q Þ n j¼1;j6¼i ðk i k j Þ : ð4þ Proof. The proof follows closely the proofs of Lemma 2.1 and Theorem 2.1 in [7]. Let Y H ; X 2 C nn be such that J ¼ Y H AX is a Jordan decomposition of A, and let B ¼ðA kiþ 1 for k 6¼ k i, where i ¼ 1;...; n. It is easy to see that B ¼ XðJ kiþ 1 Y H, and thus
4 168 O. Ovaskainen / Mathematical Biosciences 181 (2003) b kl ¼ Xn x ðjþ k yðjþ l k j¼1 j k ; where b kl is the ðk; lþ element of B. Ask i is a simple eigenvalue, it follows that limðk i kþb kl ¼ x ðiþ k k!k yðiþ l : i ð5þ ð6þ Let e d denote the dth unit vector with e d i ¼ d di, and let ða kiþ d denote the matrix A ki with its dth column replaced by the unit vector e d. By Eq. (6) and by CramerÕs rule, x ðiþ d yðiþ d ¼ limðk i kþ det½ða kiþ d Š k!ki detða kiþ ; Q n 1 x ðiþ d yðiþ j¼1 d ¼ limðk k i Þ ðk k j Q Þ n k!ki j¼1 ðk k jþ ; ð7þ ð8þ from which Eq. (3) follows. Eq. (4) follows in a similar fashion. I will next derive two eigenvalue perturbation formulae approximating the change in a simple eigenvalue due to rank-one modifications. The formulae will appear as corollaries to Theorem 2.1. I will denote by mdða; e AÞ the matching distance between the eigenvalues of A and e A, defined by mdða; AÞ¼min e fmax j k ~ pðiþ k i jg; p i where p is taken over all permutations of f1; 2;...; ng. Corollary 2.1. Let A 2 C nn, and let A 2 C nn be the matrix obtained from A by setting the elements in the dth row and dth column to zero. Let k be a simple eigenvalue of A with corresponding right and left eigenvectors x and y (normalized as y H x ¼ 1). Let d be the minimum distance between k and the other eigenvalues of A, and let m ¼ mdða; A Þ: Then, if m < d, A has a unique eigenvalue k for which k k ¼ ky d x d ð1 þ Þ; ð10þ where jj 6 n 1 d 1 ¼ d m ðn 1Þm þ Oðm 2 Þ: d Proof. Let k i and k i denote the eigenvalues of A and A, arranged so that k 1 ¼ k and k i matches k i in the sense of Eq. (15). By Theorem 2.1, Y n k k k k j ¼ ky d x d k k : ð12þ j¼2 j The claim follows as ðk k j Þ=ðk k j Þ¼1þ j, where j j j < m=ðd mþ. ð9þ ð11þ
5 O. Ovaskainen / Mathematical Biosciences 181 (2003) Corollary 2.2. Let A 2 C nn, and let A; e A þ 2 C ðnþ1þðnþ1þ be the matrices ea ¼ A 0 ; A þ ¼ A a C 0 0 a H : R a Let k 6¼ 0 be a simple eigenvalue of A with corresponding right and left eigenvectors x and y (normalized as y H x ¼ 1). Let d be the minimum distance between k and the other eigenvalues of A augmented by zero, and let m ¼ mdða; A þ Þ: Then, if m < d, A þ has an unique eigenvalue k þ for which k þ k ¼ 1 k ðyðiþh a C Þða H R xðiþ Þð1 þ Þ; ð13þ where n d jj 6 1 ¼ nm d m d þ Oðm2 Þ: Proof. Proof is similar to that of Corollary 2.1. ð14þ In order to use Corollaries 2.1 and 2.2, one should be able to say that the matching distance mdða; AÞ e (where A e ¼ A or A e ¼ A þ ) is sufficiently small. It is well-known that this is guaranteed if the matrix E ¼ A e A is sufficiently small. In the general case, it holds that [10] mdða; AÞ e 6 ð2n 1ÞðkAk 2 þkak e 2 Þ 1 1=n kek 1=n 2 : ð15þ If A is diagonalizable, the bound can be improved to [10] mdða; AÞ e 6 ð2n 1ÞkX 1 EX k; ð16þ where X 1 AX ¼ K is a diagonalization of A and kk denotes any consistent matrix norm such that kdiagða 1 ;...; a n Þk ¼ max i ja i j. For example, kkmay be any matrix norm subordinate to an absolute vector norm [10]. 3. Habitat destruction and habitat restoration In this section I will apply the results derived in Section 2 to analyze the consequences of habitat destruction and habitat restoration in the metapopulation context. Consider a species inhabiting a highly fragmented landscape consisting of a discrete set of n habitat patches (for a thorough discussion of the ecology of such species see [11]). According to the metapopulation concept, the species may persist in the network if local extinctions are compensated for by recolonizations of empty habitat patches. For the purpose of illustration, I model the colonization extinction dynamics of the species with a relatively simple metapopulation model, the spatially realistic version of the Levins model [6], which belongs to the larger family of patch occupancy models [12]. The spatially realistic Levins model is a deterministic continuoustime model, which gives the rate of change in the probability p i of patch i being occupied as [6,13]
6 170 O. Ovaskainen / Mathematical Biosciences 181 (2003) dp i dt ¼ C ið1 p i Þ E i p i ; ð17þ where C i and E i are the colonization and extinction rates of patch i, respectively. Let A i denote the area of patch i and d ij the distance between patches i and j. I assume that C i ¼ ca f im i Pj6¼i Af em j f ðd ij Þp j and that E i ¼ e=a f ex i. Here e and c are extinction and colonization rate parameters, and f im, f em and f ex describe the scaling of immigration, emigration and extinction rates by patch area, respectively. The justification of the functional form for C i is based on the reasoning that the colonization rate of an empty patch increases with increasing size of the patch, with increasing number of potential source patches (occupied patches in the vicinity of the empty patch), with increasing sizes of potential source patches, and with decreasing distances to potential source patches. The function f describes the dispersal kernel, which gives the effect of distance on colonization success. For example, [6] made the phenomenological assumption that f ðdþ ¼e ad with 1=a giving the average migration distance. More generally, the function f may be derived from submodels (such as correlated random walks) for the migration phase. The functional form for the extinction rate E i is based on the reasoning that large patches tend to have large population sizes, and they consequently have a small extinction risk. For further discussion and justification of the functional forms of the colonization and extinction processes see [11,12]. The above assumptions lead to a Ôlandscape matrixõ M with elements m ij ¼ Af exþf im i A f em j f ðd ij Þ for j 6¼ i; ð18þ 0 for j ¼ i: The element ðc=eþm ij gives the contribution that patch j makes to the colonization rate of patch i when patch i is empty, multiplied by the expected lifetime of patch i when the patch is occupied. As the fraction of time that patch i is occupied is determined by its colonization and extinction rates, ðc=eþm ij may be viewed to measure the fraction of time that patch i would be occupied if the only source of immigrants would be patch j. The metapopulation capacity k M of the habitat patch network is defined as the leading eigenvalue of matrix M. Defining d ¼ e=c, the threshold condition for persistence (in the sense of the existence of a stable non-trivial equilibrium state of Eq. (17)) is given as [6,12] k M > d: ð19þ In Eq. (19), the metapopulation capacity k M is a landscape index measuring the capacity of the habitat patch network to support the long-term persistence of a species, whereas the species parameter d sets the threshold value for the persistence of the particular species. The parameter d is independent of the landscape, 1=d measuring the expected number of colonization events that a local population inhabiting a unit size patch would cause in its lifetime to an empty patch of unit size located at distance d with f ðdþ ¼1. Metapopulation capacity is analogous to the basic reproduction ratio R 0, which has been used in epidemiology to describe the number of new infections that a single infectious individual is expected to give raise to [2]. The two quantities are related by R 0 ¼ k M =d, and thus the well-known threshold condition R 0 > 1 is equivalent with the condition given by Eq. (19). The reason for separating the species parameter d in the metapopulation context is that Eq. (19) allows one to analyse the metapopulation capacity of a landscape even for species for which the parameter d is not known.
7 O. Ovaskainen / Mathematical Biosciences 181 (2003) While metapopulation capacity characterizes the capacity of the entire network to support the persistence of a species, it would often be desirable to be able to assess the values of individual habitat patches. The ÔvalueÕ of a habitat patch is not a well-defined quantity as such, as a multitude of measures can be justified. For example, one might consider the contribution of a habitat patch to the expected size of the metapopulation, or the contribution of a habitat patch to the expected time to metapopulation extinction [14,15]. I will concentrate here on a simple measure that is appropriate for the case of rare species, which is V i, the contribution of a habitat patch i to the metapopulation capacity of the landscape. More precisely, V i is defined as the decrease in metapopulation capacity due to the destruction of patch i. As the destruction of patch i corresponds to the deletion of a row and a column from matrix M, and as the leading eigenvalue of an irreducible non-negative matrix is simple, the theory developed in Section 2 applies immediately. Most importantly, Corollary 2.1 gives the approximation formula V i ev i ¼ k M y i x i ; ð20þ where x and y are the right and the left leading eigenvectors corresponding to the eigenvalue k M. Similarly, one may apply Corollary 2.2 to consider the value of a new patch that would be added to the network, by analyzing the increase in metapopulation capacity due to the addition of a new row and a new column to matrix M. For the purpose of illustration, consider the hypothetical network of 30 habitat patches shown in Fig. 1. Fig. 2 compares the approximations ev i for all the 30 patches with the exact values. The approximations are very close to the exact values, the largest relative deviations occurring for the patches that have the largest contribution to the metapopulation capacity of the patch network, as expected from Corollary 2.1. Fig. 1. A hypothetical habitat patch network of 30 patches used in Figs The patches are randomly located within a5 5 square, patch areas being lognormally distributed with mean 1 and standard deviation 1. The patch indicated by an arrow is further analyzed in Fig. 3. The contour lines depict the value of the function gðzþ in Eq. (24). Moving one contour line away from the core of the network corresponds to the reduction of the value of gðzþ by one half.
8 172 O. Ovaskainen / Mathematical Biosciences 181 (2003) Fig. 2. The accuracy of the approximation formula (20). The dots depict the true and approximate contributions of individual habitat patches to the metapopulation capacity of the patch network shown in Fig. 1. The line shows identity. The parameter values f ex ¼ 0:8, f im ¼ 0:5, f em ¼ 0:5 and f ðdþ ¼e d are used also in Figs. 3 and 4. The approximation formulae (Corollaries 2.1 and 2.2) relate in an interesting way to the traditional eigenvalue perturbation theory. It is well known that if matrix A is perturbed to a matrix ea as e A ¼ A þ E, a simple eigenvalue k of A is perturbed to an eigenvalue ~ k of e A as [10,16] ~k ¼ k þ y H Ex þ OðkEk 2 Þ; ð21þ where x and y are the right and the left eigenvectors (normalized as y H x ¼ 1) corresponding to k. Comparison of Eqs. (21) and (10) reveals that the metapopulation capacity of the habitat patch network behaves non-linearly with respect to gradual habitat deterioration. This is illustrated in Fig. 3, where I consider a perturbation obtained by multiplying the area of a single patch by 1 s, where 0 < s < 1 represents the loss of patch area. Applying Eq. (21) for small s and extrapolating to s ¼ 1 would lead to V i fk M y i x i ; ð22þ where f ¼ f im þ f em þ f ex is called the patch area scaling factor of the model [15]. The Eq. (22) is, in the general case, in disagreement with Eq. (20). The problem arises because the error term OðkEk 2 Þ in Eq. (21) is of the same magnitude as the first order term ðy H ExÞ=y H x for s ¼ 1. In Fig. 3a, f ¼ 1:8 > 1, and consequently habitat deterioration is most detrimental in the early stage of destruction when the patch loosing area is still large. On the contrary, in Fig. 3b, where f ¼ 0:5 < 1, habitat deterioration is most detrimental at the stage when most of the patch has already been destroyed. Corollaries 2.1 and 2.2 together with Eqs. (15) and (16) give explicit upper bounds for the error in the approximations, the convergence rate ¼ OðkEkÞ being optimal for the case of diagonalizable matrices. A caveat to the error bounds is that the quantitative bounds are likely to largely
9 O. Ovaskainen / Mathematical Biosciences 181 (2003) Fig. 3. The effect of gradual deterioration (loss of area) of the patch indicated by an arrow in Fig. 1. The continuous curves depict the change in the metapopulation capacity k M of the patch network due to deterioration of the patch obtained by multiplying the patch area by the factor 1 s. Dot A depicts the metapopulation capacity of the original patch network, and dot B gives the metapopulation capacity of the patch network from which the patch has been removed. Dots C and D are approximations of dot B, dot C being based on the approximation formula (20), and dot D being based on linear extrapolation of the derivative as given by Eq. (22). In panel (a), the parameter values are as in Fig. 2, whereas in panel (b) they have been modified to f ex ¼ 0:3, f im ¼ 0:1, f em ¼ 0:1. overestimate the true errors. Fig. 4 gives an example, panel (a) showing the error bars given by Corollary 2.1 together with Eq. (16) for the example in Fig. 2. The error bars become narrower only when the relative contributions of the habitat patches to the metapopulation capacity are negligible (Fig. 4b). Corollary 2.2 allows us to analyze the effect of habitat restoration, defined here as a creation of a new patch in a particular location. Consider again a network of n patches, and let a new patch of area A be added to the network in a location with distances d i (i ¼ 1;...; n) from the existing patches. Utilizing Corollary 2.2, the increase in metapopulation capacity due to the addition of the new patch is given by!! X n k þ M k M Af k M X n i¼1 y i A f exþf im i f ðd i Þ i¼1 x i A f em i f ðd i Þ ; ð23þ
10 174 O. Ovaskainen / Mathematical Biosciences 181 (2003) Fig. 4. The error bounds given by Corollary 2.1 together with Eq. (16). Panel (a) shows the error bars for the example shown in Fig. 2. In panel (b), the structure of the habitat patch network has been modified so that the patch areas are lognormally distributed with mean 1 and standard deviation 4. The line shows identity. where the accuracy of the approximation increases with decreasing size of the new patch. A convenient feature of Eq. (23) is that it separates the effects of area and location of the patch to be added to the network. Letting z ¼ðz x ; z y Þ denote the ðx; yþ-coordinates of the new patch, Eq. (23) suggests that the value of the location z is given by!! X n gðzþ ¼ Xn i¼1 y i A f exþf im i f ðd i Þ i¼1 x i A f em i f ðd i Þ ; ð24þ where the right-hand side depends on the location z through the distances d i. The function gðzþ reveals the core of the habitat patch network, as illustrated by the contour lines in Fig. 1. By Eq. (23), the effect of the area of the new patch to be added to the network scales to the power of f. As expected, the more sensitive the basic ingredients of metapopulation dynamics (local extinction, immigration and emigration) are to patch area, the more important is the area of the patch to be restored.
11 O. Ovaskainen / Mathematical Biosciences 181 (2003) Discussion The main contribution of this paper is to generalize eigenvalue eigenvector relations for nonsymmetric matrices. While the exact eigenvalue eigenvector relations given by Theorem 2.1 are mainly of theoretical interest, the approximations given by Corollaries 2.1 and 2.2 can be readily applied to practical problems. As illustrated by Fig. 2, the approximations may be fairly accurate for a number of potential applications, while the quantitative error bounds may often be far too pessimistic (Fig. 4). However, one should note that the error bounds apply to the general case, whereas Fig. 4 relates to the approximation of the leading eigenvalue of a non-negative matrix, which is certainly a special case. In order to demonstrate the applicability of eigenvalue eigenvector relations in population biological theory, I have applied Corollaries 2.1 and 2.2 to examine the effects of habitat destruction and habitat restoration in the metapopulation context. This problem has been studied earlier mainly through simulation studies (see e.g. [17,18]), where it is possible to include a much more detailed description of the ecology of the species, but at the same time the analysis is restricted to comparing numerical results for a set of parameter values. Analytical results from simpler models such as the one studied in this paper enhance general understanding and bring conceptual clarity to the interpretation of simulation results. For example, it would have been hard to demonstrate with a numerical study the separation of the contributions of patch area and patch location to metapopulation capacity (Eq. (23) and the contour lines in Fig. 1). Similarly, it would be hard to see with numerical studies only the effect that f has on the non-linear relationships between gradual habitat deterioration and the metapopulation capacity of a patch network (Eqs. (20) and (22) and Fig. 3). As the mathematical theory presented in Section 2 is fairly general, I expect that its applicability is not restricted to the metapopulation context. As an example, one potential application of these results might be found in epidemiology, where the basic reproduction ratio R 0 is given by the leading eigenvalue of a matrix describing the spatial or social heterogeneity in the host population [2]. Vaccination scenarios effectively correspond to removal of susceptibles, their effect being thus measured by the decrease in R 0 due to rank modifications of the relevant matrix. Another application could be found in the study of food webs (see e.g. [3,19]), where the effect of extinction or introduction of species on community stability may be measured in the change of the eigenvalues of the community matrix due to rank modifications. Indeed, as the removal (or addition) of matrix rows and columns often corresponds in a natural way to decreasing (or increasing) the degrees of freedom of the focal system, and as eigenvalues are of major importance in characterizing the behaviour of systems described through matrices, useful applications of the theory presented here should be found also in other fields of biology, engineering and other applied sciences. Acknowledgements I thank Ilkka Hanski and Marko Huhtanen for their valuable comments. This study was supported by the Academy of Finland (grant number and the Finnish Centre of Excellence Programme , grant number 44887).
12 176 O. Ovaskainen / Mathematical Biosciences 181 (2003) References [1] H. Caswell, Matrix population models. Construction, Analysis, and Interpretation, Sinauer Associates, Sunderland, MA, [2] O. Diekmann, J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, John Wiley, Chichester, [3] J. Jorgensen, A.M. Rossignol, C.J. Puccia, R. Levins, P.A. Rossignol, On the variance of eigenvalues of the community matrix: derivation and appraisal, Ecology 81 (2000) [4] J.B. McGraw, H. Caswell, Estimation of individual fitness from life-history data, Am. Nat. 147 (1996) 47. [5] M.C. Whitlock, N.H. Barton, The effective size of a subdivided population, Genetics 146 (1997) 427. [6] I. Hanski, O. Ovaskainen, The metapopulation capacity of a fragmented landscape, Nature 404 (2000) 755. [7] S. Elhay, G.M.L. Gladwell, G.H. Golub, Y.M. Ram, On some eigenvector eigenvalue relations, SIAM J. Matrix Anal. Appl. 20 (1999) 563. [8] G.H. Golub, Some uses of the Lanczos algorith in numerical linear algebra, in: J.H.H. Miller (Ed.), Topics in Numerical Analysis, Springer, Heidelberg, 1973, p. 23. [9] Y.M. Ram, Inverse eigenvalue problems for a modified vibrating system, SIAM J. Appl. Math. 53 (1993) [10] G.W. Stewart, J. Sun, Matrix Perturbation Theory, Academic Press, San Diego, CA, [11] I. Hanski, Metapopulation Ecology, Oxford University, Oxford, [12] O. Ovaskainen, I. Hanski, Spatially structured metapopulation models: global and local assessment of metapopulation capacity, Theor. Popul. Biol. 60 (2001) 281. [13] I. Hanski, M. Gyllenberg, Uniting two general patterns in the distribution of species, Science 275 (1997) 397. [14] R.S. Etienne, J.A.P. Heesterbeek, Rules of thumb for conservation of metapopulations based on a stochastic winking-patch model, Am. Nat. 158 (2001) 389. [15] O. Ovaskainen, I. Hanski, How much does an individual habitat fragment contribute to metapopulation dynamics and persistence? Manuscript. [16] G.H. Golub, C.E. van Loan, Matrix Computations, Johns Hopkins University, Baltimore, MD, [17] D.B. Lindenmayer, H.O. Possingham, Modelling the inter-relationships between habitat patchiness, dispersal capability and metapopulation persistence of the endangered species LeadbeaterÕs possum, in south-eastern Australia, Landsc. Ecol. 11 (1996) 79. [18] P. Gaona, P. Ferreras, M. Delibes, Dynamics and viability of a metapopulation of the endangered Iberian lynx (Lynx pardinus), Ecol. Monog. 68 (1998) 349. [19] K. McCann, A. Hastings, G.R. Huxel, Weak interactions and the balance of nature, Nature 395 (1998) 794.
Extinction threshold in metapopulation models
Ann. Zool. Fennici 40: 8 97 ISSN 0003-455X Helsinki 30 April 2003 Finnish Zoological and Botanical Publishing Board 2003 Extinction threshold in metapopulation models Otso Ovaskainen & Ilkka Hanski Metapopulation
More informationChapter 5 Lecture. Metapopulation Ecology. Spring 2013
Chapter 5 Lecture Metapopulation Ecology Spring 2013 5.1 Fundamentals of Metapopulation Ecology Populations have a spatial component and their persistence is based upon: Gene flow ~ immigrations and emigrations
More informationMetapopulation modeling: Stochastic Patch Occupancy Model (SPOM) by Atte Moilanen
Metapopulation modeling: Stochastic Patch Occupancy Model (SPOM) by Atte Moilanen 1. Metapopulation processes and variables 2. Stochastic Patch Occupancy Models (SPOMs) 3. Connectivity in metapopulation
More informationMerging Spatial and Temporal Structure within a Metapopulation Model
Merging within a Metapopulation Model manuscript Yssa D. DeWoody Department of Forestry and Natural Resources, Purdue University, West Lafayette, Indiana 47907-2033; (765) 494-3604; (765) 496-2422 (fax);
More informationHabitat Deterioration, Habitat Destruction, and Metapopulation Persistence in a Heterogenous Landscape
Theoretical Population Biology 52, 198215 (1997) Article No. TP971333 Habitat Deterioration, Habitat Destruction, and Metapopulation Persistence in a Heterogenous Landscape Mats Gyllenberg Department of
More informationFast principal component analysis using fixed-point algorithm
Pattern Recognition Letters 28 (27) 1151 1155 www.elsevier.com/locate/patrec Fast principal component analysis using fixed-point algorithm Alok Sharma *, Kuldip K. Paliwal Signal Processing Lab, Griffith
More informationEcology Regulation, Fluctuations and Metapopulations
Ecology Regulation, Fluctuations and Metapopulations The Influence of Density on Population Growth and Consideration of Geographic Structure in Populations Predictions of Logistic Growth The reality of
More information6 Metapopulations of Butterflies (sketch of the chapter)
6 Metapopulations of Butterflies (sketch of the chapter) Butterflies inhabit an unpredictable world. Consider the checkerspot butterfly, Melitaea cinxia, also known as the Glanville Fritillary. They depend
More informationThe effect of emigration and immigration on the dynamics of a discrete-generation population
J. Biosci., Vol. 20. Number 3, June 1995, pp 397 407. Printed in India. The effect of emigration and immigration on the dynamics of a discrete-generation population G D RUXTON Biomathematics and Statistics
More informationA note on variational representation for singular values of matrix
Alied Mathematics and Comutation 43 (2003) 559 563 www.elsevier.com/locate/amc A note on variational reresentation for singular values of matrix Zhi-Hao Cao *, Li-Hong Feng Deartment of Mathematics and
More informationThe Second Eigenvalue of the Google Matrix
The Second Eigenvalue of the Google Matrix Taher H. Haveliwala and Sepandar D. Kamvar Stanford University {taherh,sdkamvar}@cs.stanford.edu Abstract. We determine analytically the modulus of the second
More informationPopulation viability analysis
Population viability analysis Introduction The process of using models to determine risks of decline faced by populations was initially defined as population vulnerability analysis [1], but is now known
More informationc 2005 Society for Industrial and Applied Mathematics
SIAM J. MATRIX ANAL. APPL. Vol. 27, No. 2, pp. 305 32 c 2005 Society for Industrial and Applied Mathematics JORDAN CANONICAL FORM OF THE GOOGLE MATRIX: A POTENTIAL CONTRIBUTION TO THE PAGERANK COMPUTATION
More informationEFFECTS OF SUCCESSIONAL DYNAMICS ON METAPOPULATION PERSISTENCE
Ecology, 84(4), 2003, pp. 882 889 2003 by the Ecological Society of America EFFECTS OF SUCCESSIONAL DYNAMICS ON METAPOPULATION PERSISTENCE STEPHEN P. ELLNER 1 AND GREGOR FUSSMANN 2 Department of Ecology
More informationThe Lanczos and conjugate gradient algorithms
The Lanczos and conjugate gradient algorithms Gérard MEURANT October, 2008 1 The Lanczos algorithm 2 The Lanczos algorithm in finite precision 3 The nonsymmetric Lanczos algorithm 4 The Golub Kahan bidiagonalization
More informationEIGENVALUES AND SINGULAR VALUE DECOMPOSITION
APPENDIX B EIGENVALUES AND SINGULAR VALUE DECOMPOSITION B.1 LINEAR EQUATIONS AND INVERSES Problems of linear estimation can be written in terms of a linear matrix equation whose solution provides the required
More informationTHE RELATION BETWEEN THE QR AND LR ALGORITHMS
SIAM J. MATRIX ANAL. APPL. c 1998 Society for Industrial and Applied Mathematics Vol. 19, No. 2, pp. 551 555, April 1998 017 THE RELATION BETWEEN THE QR AND LR ALGORITHMS HONGGUO XU Abstract. For an Hermitian
More informationYimin Wei a,b,,1, Xiezhang Li c,2, Fanbin Bu d, Fuzhen Zhang e. Abstract
Linear Algebra and its Applications 49 (006) 765 77 wwwelseviercom/locate/laa Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices Application of perturbation theory
More informationComparison of perturbation bounds for the stationary distribution of a Markov chain
Linear Algebra and its Applications 335 (00) 37 50 www.elsevier.com/locate/laa Comparison of perturbation bounds for the stationary distribution of a Markov chain Grace E. Cho a, Carl D. Meyer b,, a Mathematics
More informationEIGENVALUES AND EIGENVECTORS 3
EIGENVALUES AND EIGENVECTORS 3 1. Motivation 1.1. Diagonal matrices. Perhaps the simplest type of linear transformations are those whose matrix is diagonal (in some basis). Consider for example the matrices
More informationCOURSE SCHEDULE. Other applications of genetics in conservation Resolving taxonomic uncertainty
Tutorials: Next week, Tues. 5 Oct. meet in from of Library Processing entre (path near Woodward) at 2pm. We re going on a walk in the woods, so dress appropriately! Following week, Tues. 2 Oct.: Global
More informationEigenvalue inverse formulation for optimising vibratory behaviour of truss and continuous structures
Computers and Structures 8 () 397 43 www.elsevier.com/locate/compstruc Eigenvalue inverse formulation for optimising vibratory behaviour of truss and continuous structures H. Bahai *, K. Farahani, M.S.
More informationSpectrally arbitrary star sign patterns
Linear Algebra and its Applications 400 (2005) 99 119 wwwelseviercom/locate/laa Spectrally arbitrary star sign patterns G MacGillivray, RM Tifenbach, P van den Driessche Department of Mathematics and Statistics,
More informationStability Of Specialists Feeding On A Generalist
Stability Of Specialists Feeding On A Generalist Tomoyuki Sakata, Kei-ichi Tainaka, Yu Ito and Jin Yoshimura Department of Systems Engineering, Shizuoka University Abstract The investigation of ecosystem
More informationThe interplay between immigration and local population dynamics in metapopulations. Ovaskainen, Otso.
https://helda.helsinki.fi The interplay between immigration and local population dynamics in metapopulations Ovaskainen, Otso 27-4 Ovaskainen, O 27, ' The interplay between immigration and local population
More informationAn algorithm for symmetric generalized inverse eigenvalue problems
Linear Algebra and its Applications 296 (999) 79±98 www.elsevier.com/locate/laa An algorithm for symmetric generalized inverse eigenvalue problems Hua Dai Department of Mathematics, Nanjing University
More informationLattice models of habitat destruction in a prey-predator system
22nd International Congress on Modelling and Simulation, Hobart, Tasmania, Australia, 3 to 8 December 2017 mssanz.org.au/modsim2017 Lattice models of habitat destruction in a prey-predator system Nariiyuki
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More informationIMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET
IMPORTANT DEFINITIONS AND THEOREMS REFERENCE SHEET This is a (not quite comprehensive) list of definitions and theorems given in Math 1553. Pay particular attention to the ones in red. Study Tip For each
More information6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities
6.207/14.15: Networks Lectures 4, 5 & 6: Linear Dynamics, Markov Chains, Centralities 1 Outline Outline Dynamical systems. Linear and Non-linear. Convergence. Linear algebra and Lyapunov functions. Markov
More informationConvergence of a linear recursive sequence
int. j. math. educ. sci. technol., 2004 vol. 35, no. 1, 51 63 Convergence of a linear recursive sequence E. G. TAY*, T. L. TOH, F. M. DONG and T. Y. LEE Mathematics and Mathematics Education, National
More informationOn the eigenvalues of specially low-rank perturbed matrices
On the eigenvalues of specially low-rank perturbed matrices Yunkai Zhou April 12, 2011 Abstract We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is
More informationTHE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR
THE PERTURBATION BOUND FOR THE SPECTRAL RADIUS OF A NON-NEGATIVE TENSOR WEN LI AND MICHAEL K. NG Abstract. In this paper, we study the perturbation bound for the spectral radius of an m th - order n-dimensional
More informationSpatial Dimensions of Population Viability
Spatial Dimensions of Population Viability Gyllenberg, M., Hanski, I. and Metz, J.A.J. IIASA Interim Report November 2004 Gyllenberg, M., Hanski, I. and Metz, J.A.J. (2004) Spatial Dimensions of Population
More informationIntrinsic products and factorizations of matrices
Available online at www.sciencedirect.com Linear Algebra and its Applications 428 (2008) 5 3 www.elsevier.com/locate/laa Intrinsic products and factorizations of matrices Miroslav Fiedler Academy of Sciences
More informationLinear Algebra. Solving Linear Systems. Copyright 2005, W.R. Winfrey
Copyright 2005, W.R. Winfrey Topics Preliminaries Echelon Form of a Matrix Elementary Matrices; Finding A -1 Equivalent Matrices LU-Factorization Topics Preliminaries Echelon Form of a Matrix Elementary
More informationMath 1553, Introduction to Linear Algebra
Learning goals articulate what students are expected to be able to do in a course that can be measured. This course has course-level learning goals that pertain to the entire course, and section-level
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2016 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationA Note on Eigenvalues of Perturbed Hermitian Matrices
A Note on Eigenvalues of Perturbed Hermitian Matrices Chi-Kwong Li Ren-Cang Li July 2004 Let ( H1 E A = E H 2 Abstract and à = ( H1 H 2 be Hermitian matrices with eigenvalues λ 1 λ k and λ 1 λ k, respectively.
More informationExample Linear Algebra Competency Test
Example Linear Algebra Competency Test The 4 questions below are a combination of True or False, multiple choice, fill in the blank, and computations involving matrices and vectors. In the latter case,
More informationj=1 u 1jv 1j. 1/ 2 Lemma 1. An orthogonal set of vectors must be linearly independent.
Lecture Notes: Orthogonal and Symmetric Matrices Yufei Tao Department of Computer Science and Engineering Chinese University of Hong Kong taoyf@cse.cuhk.edu.hk Orthogonal Matrix Definition. Let u = [u
More informationSolution of the Inverse Eigenvalue Problem for Certain (Anti-) Hermitian Matrices Using Newton s Method
Journal of Mathematics Research; Vol 6, No ; 014 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education Solution of the Inverse Eigenvalue Problem for Certain (Anti-) Hermitian
More informationSolution: a) Let us consider the matrix form of the system, focusing on the augmented matrix: 0 k k + 1. k 2 1 = 0. k = 1
Exercise. Given the system of equations 8 < : x + y + z x + y + z k x + k y + z k where k R. a) Study the system in terms of the values of k. b) Find the solutions when k, using the reduced row echelon
More informationThe inverse of a tridiagonal matrix
Linear Algebra and its Applications 325 (2001) 109 139 www.elsevier.com/locate/laa The inverse of a tridiagonal matrix Ranjan K. Mallik Department of Electrical Engineering, Indian Institute of Technology,
More informationWe first repeat some well known facts about condition numbers for normwise and componentwise perturbations. Consider the matrix
BIT 39(1), pp. 143 151, 1999 ILL-CONDITIONEDNESS NEEDS NOT BE COMPONENTWISE NEAR TO ILL-POSEDNESS FOR LEAST SQUARES PROBLEMS SIEGFRIED M. RUMP Abstract. The condition number of a problem measures the sensitivity
More informationDifferentiation matrices in polynomial bases
Math Sci () 5 DOI /s9---x ORIGINAL RESEARCH Differentiation matrices in polynomial bases A Amiraslani Received January 5 / Accepted April / Published online April The Author(s) This article is published
More informationLimit theorems for discrete-time metapopulation models
MASCOS AustMS Meeting, October 2008 - Page 1 Limit theorems for discrete-time metapopulation models Phil Pollett Department of Mathematics The University of Queensland http://www.maths.uq.edu.au/ pkp AUSTRALIAN
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 14 (2009) 4041 4056 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns Symbolic computation
More informationHabitat fragmentation and evolution of dispersal. Jean-François Le Galliard CNRS, University of Paris 6, France
Habitat fragmentation and evolution of dispersal Jean-François Le Galliard CNRS, University of Paris 6, France Habitat fragmentation : facts Habitat fragmentation describes a state (or a process) of discontinuities
More informationSome inequalities for sum and product of positive semide nite matrices
Linear Algebra and its Applications 293 (1999) 39±49 www.elsevier.com/locate/laa Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b,
More informationBehaviour of simple population models under ecological processes
J. Biosci., Vol. 19, Number 2, June 1994, pp 247 254. Printed in India. Behaviour of simple population models under ecological processes SOMDATTA SINHA* and S PARTHASARATHY Centre for Cellular and Molecular
More informationSENSITIVITY OF THE STATIONARY DISTRIBUTION OF A MARKOV CHAIN*
SIAM J Matrix Anal Appl c 1994 Society for Industrial and Applied Mathematics Vol 15, No 3, pp 715-728, July, 1994 001 SENSITIVITY OF THE STATIONARY DISTRIBUTION OF A MARKOV CHAIN* CARL D MEYER Abstract
More informationPattern correlation matrices and their properties
Linear Algebra and its Applications 327 (2001) 105 114 www.elsevier.com/locate/laa Pattern correlation matrices and their properties Andrew L. Rukhin Department of Mathematics and Statistics, University
More informationTopic 1: Matrix diagonalization
Topic : Matrix diagonalization Review of Matrices and Determinants Definition A matrix is a rectangular array of real numbers a a a m a A = a a m a n a n a nm The matrix is said to be of order n m if it
More informationMATRICES. knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns.
MATRICES After studying this chapter you will acquire the skills in knowledge on matrices Knowledge on matrix operations. Matrix as a tool of solving linear equations with two or three unknowns. List of
More informationLinear Algebra. Min Yan
Linear Algebra Min Yan January 2, 2018 2 Contents 1 Vector Space 7 1.1 Definition................................. 7 1.1.1 Axioms of Vector Space..................... 7 1.1.2 Consequence of Axiom......................
More informationOptimal Translocation Strategies for Threatened Species
Optimal Translocation Strategies for Threatened Species Rout, T. M., C. E. Hauser and H. P. Possingham The Ecology Centre, University of Queensland, E-Mail: s428598@student.uq.edu.au Keywords: threatened
More informationHW2 - Due 01/30. Each answer must be mathematically justified. Don t forget your name.
HW2 - Due 0/30 Each answer must be mathematically justified. Don t forget your name. Problem. Use the row reduction algorithm to find the inverse of the matrix 0 0, 2 3 5 if it exists. Double check your
More informationLinear Algebra review Powers of a diagonalizable matrix Spectral decomposition
Linear Algebra review Powers of a diagonalizable matrix Spectral decomposition Prof. Tesler Math 283 Fall 2018 Also see the separate version of this with Matlab and R commands. Prof. Tesler Diagonalizing
More informationDoes spatial structure facilitate coexistence of identical competitors?
Ecological Modelling 181 2005 17 23 Does spatial structure facilitate coexistence of identical competitors? Zong-Ling Wang a, Da-Yong Zhang b,, Gang Wang c a First Institute of Oceanography, State Oceanic
More informationNatal versus breeding dispersal: Evolution in a model system
Evolutionary Ecology Research, 1999, 1: 911 921 Natal versus breeding dispersal: Evolution in a model system Karin Johst 1 * and Roland Brandl 2 1 Centre for Environmental Research Leipzig-Halle Ltd, Department
More informationA Tutorial on Data Reduction. Principal Component Analysis Theoretical Discussion. By Shireen Elhabian and Aly Farag
A Tutorial on Data Reduction Principal Component Analysis Theoretical Discussion By Shireen Elhabian and Aly Farag University of Louisville, CVIP Lab November 2008 PCA PCA is A backbone of modern data
More informationCentral Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J
Central Groupoids, Central Digraphs, and Zero-One Matrices A Satisfying A 2 = J Frank Curtis, John Drew, Chi-Kwong Li, and Daniel Pragel September 25, 2003 Abstract We study central groupoids, central
More informationON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH
ON ORTHOGONAL REDUCTION TO HESSENBERG FORM WITH SMALL BANDWIDTH V. FABER, J. LIESEN, AND P. TICHÝ Abstract. Numerous algorithms in numerical linear algebra are based on the reduction of a given matrix
More informationImproved Newton s method with exact line searches to solve quadratic matrix equation
Journal of Computational and Applied Mathematics 222 (2008) 645 654 wwwelseviercom/locate/cam Improved Newton s method with exact line searches to solve quadratic matrix equation Jian-hui Long, Xi-yan
More informationMarkov Chains, Stochastic Processes, and Matrix Decompositions
Markov Chains, Stochastic Processes, and Matrix Decompositions 5 May 2014 Outline 1 Markov Chains Outline 1 Markov Chains 2 Introduction Perron-Frobenius Matrix Decompositions and Markov Chains Spectral
More informationThe dynamics of disease transmission in a Prey Predator System with harvesting of prey
ISSN: 78 Volume, Issue, April The dynamics of disease transmission in a Prey Predator System with harvesting of prey, Kul Bhushan Agnihotri* Department of Applied Sciences and Humanties Shaheed Bhagat
More informationCHAPTER 1 - INTRODUCTION. Habitat fragmentation, or the subdivision of once-continuous tracts of habitat into
CHAPTER 1 - INTRODUCTION Habitat fragmentation, or the subdivision of once-continuous tracts of habitat into discontinuous patches, has been implicated as a primary factor in the loss of species (Harris
More informationWhat is A + B? What is A B? What is AB? What is BA? What is A 2? and B = QUESTION 2. What is the reduced row echelon matrix of A =
STUDENT S COMPANIONS IN BASIC MATH: THE ELEVENTH Matrix Reloaded by Block Buster Presumably you know the first part of matrix story, including its basic operations (addition and multiplication) and row
More informationMath 314H Solutions to Homework # 3
Math 34H Solutions to Homework # 3 Complete the exercises from the second maple assignment which can be downloaded from my linear algebra course web page Attach printouts of your work on this problem to
More informationBOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION
K Y BERNETIKA VOLUM E 46 ( 2010), NUMBER 4, P AGES 655 664 BOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION Guang-Da Hu and Qiao Zhu This paper is concerned with bounds of eigenvalues of a complex
More informationPriority areas for grizzly bear conservation in western North America: an analysis of habitat and population viability INTRODUCTION METHODS
Priority areas for grizzly bear conservation in western North America: an analysis of habitat and population viability. Carroll, C. 2005. Klamath Center for Conservation Research, Orleans, CA. Revised
More informationA Note on Simple Nonzero Finite Generalized Singular Values
A Note on Simple Nonzero Finite Generalized Singular Values Wei Ma Zheng-Jian Bai December 21 212 Abstract In this paper we study the sensitivity and second order perturbation expansions of simple nonzero
More informationReduction to the associated homogeneous system via a particular solution
June PURDUE UNIVERSITY Study Guide for the Credit Exam in (MA 5) Linear Algebra This study guide describes briefly the course materials to be covered in MA 5. In order to be qualified for the credit, one
More informationThe Singular Value Decomposition (SVD) and Principal Component Analysis (PCA)
Chapter 5 The Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) 5.1 Basics of SVD 5.1.1 Review of Key Concepts We review some key definitions and results about matrices that will
More informationMathematics. EC / EE / IN / ME / CE. for
Mathematics for EC / EE / IN / ME / CE By www.thegateacademy.com Syllabus Syllabus for Mathematics Linear Algebra: Matrix Algebra, Systems of Linear Equations, Eigenvalues and Eigenvectors. Probability
More informationMarkov Chains and Stochastic Sampling
Part I Markov Chains and Stochastic Sampling 1 Markov Chains and Random Walks on Graphs 1.1 Structure of Finite Markov Chains We shall only consider Markov chains with a finite, but usually very large,
More informationthe Unitary Polar Factor æ Ren-Cang Li P.O. Box 2008, Bldg 6012
Relative Perturbation Bounds for the Unitary Polar actor Ren-Cang Li Mathematical Science Section Oak Ridge National Laboratory P.O. Box 2008, Bldg 602 Oak Ridge, TN 3783-6367 èli@msr.epm.ornl.govè LAPACK
More informationChapter Two Elements of Linear Algebra
Chapter Two Elements of Linear Algebra Previously, in chapter one, we have considered single first order differential equations involving a single unknown function. In the next chapter we will begin to
More informationSAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to Linear Algebra
SAMPLE OF THE STUDY MATERIAL PART OF CHAPTER 1 Introduction to 1.1. Introduction Linear algebra is a specific branch of mathematics dealing with the study of vectors, vector spaces with functions that
More informationFundamentals of Engineering Analysis (650163)
Philadelphia University Faculty of Engineering Communications and Electronics Engineering Fundamentals of Engineering Analysis (6563) Part Dr. Omar R Daoud Matrices: Introduction DEFINITION A matrix is
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationc c c c c c c c c c a 3x3 matrix C= has a determinant determined by
Linear Algebra Determinants and Eigenvalues Introduction: Many important geometric and algebraic properties of square matrices are associated with a single real number revealed by what s known as the determinant.
More informationCS 246 Review of Linear Algebra 01/17/19
1 Linear algebra In this section we will discuss vectors and matrices. We denote the (i, j)th entry of a matrix A as A ij, and the ith entry of a vector as v i. 1.1 Vectors and vector operations A vector
More informationDeterminants. Chia-Ping Chen. Linear Algebra. Professor Department of Computer Science and Engineering National Sun Yat-sen University 1/40
1/40 Determinants Chia-Ping Chen Professor Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra About Determinant A scalar function on the set of square matrices
More information2 Discrete-Time Markov Chains
2 Discrete-Time Markov Chains Angela Peace Biomathematics II MATH 5355 Spring 2017 Lecture notes follow: Allen, Linda JS. An introduction to stochastic processes with applications to biology. CRC Press,
More informationTwo Results About The Matrix Exponential
Two Results About The Matrix Exponential Hongguo Xu Abstract Two results about the matrix exponential are given. One is to characterize the matrices A which satisfy e A e AH = e AH e A, another is about
More informationMATH 425-Spring 2010 HOMEWORK ASSIGNMENTS
MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS Instructor: Shmuel Friedland Department of Mathematics, Statistics and Computer Science email: friedlan@uic.edu Last update April 18, 2010 1 HOMEWORK ASSIGNMENT
More information1 Multiply Eq. E i by λ 0: (λe i ) (E i ) 2 Multiply Eq. E j by λ and add to Eq. E i : (E i + λe j ) (E i )
Direct Methods for Linear Systems Chapter Direct Methods for Solving Linear Systems Per-Olof Persson persson@berkeleyedu Department of Mathematics University of California, Berkeley Math 18A Numerical
More informationCoexistence of competitors in deterministic and stochastic patchy environments
0.8Copyedited by: AA 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Journal of Biological Dynamics Vol. 00,
More information2. Every linear system with the same number of equations as unknowns has a unique solution.
1. For matrices A, B, C, A + B = A + C if and only if A = B. 2. Every linear system with the same number of equations as unknowns has a unique solution. 3. Every linear system with the same number of equations
More informationPreventive behavioural responses and information dissemination in network epidemic models
PROCEEDINGS OF THE XXIV CONGRESS ON DIFFERENTIAL EQUATIONS AND APPLICATIONS XIV CONGRESS ON APPLIED MATHEMATICS Cádiz, June 8-12, 215, pp. 111 115 Preventive behavioural responses and information dissemination
More informationReview of similarity transformation and Singular Value Decomposition
Review of similarity transformation and Singular Value Decomposition Nasser M Abbasi Applied Mathematics Department, California State University, Fullerton July 8 7 page compiled on June 9, 5 at 9:5pm
More informationa s 1.3 Matrix Multiplication. Know how to multiply two matrices and be able to write down the formula
Syllabus for Math 308, Paul Smith Book: Kolman-Hill Chapter 1. Linear Equations and Matrices 1.1 Systems of Linear Equations Definition of a linear equation and a solution to a linear equations. Meaning
More informationDense LU factorization and its error analysis
Dense LU factorization and its error analysis Laura Grigori INRIA and LJLL, UPMC February 2016 Plan Basis of floating point arithmetic and stability analysis Notation, results, proofs taken from [N.J.Higham,
More informationThe Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
Applied Mathematics, 05, 6, 665-675 Published Online September 05 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/046/am056048 The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
More informationMath 60. Rumbos Spring Solutions to Assignment #17
Math 60. Rumbos Spring 2009 1 Solutions to Assignment #17 a b 1. Prove that if ad bc 0 then the matrix A = is invertible and c d compute A 1. a b Solution: Let A = and assume that ad bc 0. c d First consider
More informationMTH 5102 Linear Algebra Practice Final Exam April 26, 2016
Name (Last name, First name): MTH 5 Linear Algebra Practice Final Exam April 6, 6 Exam Instructions: You have hours to complete the exam. There are a total of 9 problems. You must show your work and write
More informationPossible numbers of ones in 0 1 matrices with a given rank
Linear and Multilinear Algebra, Vol, No, 00, Possible numbers of ones in 0 1 matrices with a given rank QI HU, YAQIN LI and XINGZHI ZHAN* Department of Mathematics, East China Normal University, Shanghai
More informationMath Final December 2006 C. Robinson
Math 285-1 Final December 2006 C. Robinson 2 5 8 5 1 2 0-1 0 1. (21 Points) The matrix A = 1 2 2 3 1 8 3 2 6 has the reduced echelon form U = 0 0 1 2 0 0 0 0 0 1. 2 6 1 0 0 0 0 0 a. Find a basis for the
More information