For a 3 3 diagonal matrix we find. Thus e 1 is a eigenvector corresponding to eigenvalue λ = a 11. Thus matrix A has eigenvalues 2 and 3.
|
|
- Bethanie Randall
- 5 years ago
- Views:
Transcription
1 Closed Leotief Model Chapter 6 Eigevalues I a closed Leotief iput-output-model cosumptio ad productio coicide, i.e. V x = x = x Is this possible for the give techology matrix V? This is a special case of a so called eigevalue problem. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues / 45 Eigevalue ad Eigevector Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues / 45 Example Eigevalue ad Eigevector A vector x R, x = 0, is called eigevector of a matrix A correspodig to eigevalue λ R, if A x = λ x The eigevalues of matrix A are all umbers λ for which a eigevector does exist. For a 3 3 diagoal matrix we fid a 0 0 a A e = 0 a 0 0 = 0 = a e 0 0 a Thus e is a eigevector correspodig to eigevalue λ = a. Aalogously we fid for a diagoal matrix A e i = a ii e i So the eigevalue of a diagoal matrix are its diagoal elemets with uit vectors e i as the correspodig eigevectors. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 3 / 45 Computatio of Eigevalues I order to fid eigevectors of matrix A we have to solve equatio A x = λx = λix (A λi)x = 0. If (A λi) is ivertible the we get x = (A λi) 0 = 0. However, x = 0 caot be a eigevector (by defiitio) ad thus λ caot be a eigevalue. Thus λ is a eigevalue of A if ad oly if (A λi) is ot ivertible, i.e., if ad oly if det(a λi) = 0 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 4 / 45 Example Eigevalues Compute the eigevalues of matrix A =. 4 We have to fid all λ R where A λi vaishes. λ det(a λi) = 4 λ = λ 5λ + 6 = 0 The roots of this quadratic equatio are λ = ad λ = 3. Thus matrix A has eigevalues ad 3. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 5 / 45 Characteristic Polyomial Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 6 / 45 Computatio of Eigevectors For a matrix A det(a λi) is a polyomial of degree i λ. It is called the characteristic polyomial of matrix A. The eigevalues are the the roots of the characteristic polyomial. For that reaso eigevalues ad eigevectors are sometimes called the characteristic roots ad characteristic vectors, resp., of A. The set of all eigevalues of A is called the spectrum of A. Spectral methods make use of eigevalues. Remark: It may happe that characteristic roots are complex (λ C). These are the called complex eigevalues. Eigevectors x correspodig to a kow eigevalue λ 0 ca be computed by solvig equatio (A λ 0 I)x = 0. Eigevectors of A = correspodig to λ = : 4 x 0 (A λ I)x = = 0 Gaussia elimiatio yields: = α ad x = α v = α for a α R \ {0} Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 7 / 45 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 8 / 45
2 Eigespace If x is a eigevector correspodig to eigevalue λ, the each multiple αx is a eigevector, too: A (αx) = α(a x) = αλ x = λ (αx) If x ad y are eigevectors correspodig to the same eigevalue λ, the x + y is a eigevector, too: A (x + y) = A x + A y = λ x + λ y = λ (x + y) The set of all eigevectors correspodig to eigevalue λ (icludig zero vector 0) is thus a subspace of R ad is called the eigespace correspodig to λ. Computer programs always retur bases of eigespaces. (Beware: Bases are ot uiquely determied!) Example Eigespace Let A =. 4 Eigevector correspodig to eigevalue λ = : v = Eigevector correspodig to eigevalue λ = 3: v = Eigevectors correspodig to eigevalue λ i are all o-vaishig multiples of v i (i.e., = 0). Computer programs retur ormalized eigevectors: v = 5 5 ad v = Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 9 / 45 Example 0 Eigevalues ad Eigevectors of A = Create the characteristic polyomial ad compute its roots: λ 0 det(a λi) = 0 3 λ = ( λ) λ (λ 5) = λ Eigevalues: λ =, λ = 0, ad λ 3 = 5. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 0 / 45 Example Eigevector(s) correspodig to eigevalue λ 3 = 5 by solvig equatio ( 5) 0 (A λ 3 I)x = 0 (3 5) 0 6 ( 5) Gaussia elimiatio yields x = Thus = α, = α, ad x = 3α for arbitrary α R \ {0}. Eigevector v 3 = (, 3, 6) t Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues / 45 Example Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues / 45 Properties of Eigevalues Eigevector correspodig to λ = : v = λ = 0: v = 6 λ 3 = 5: v 3 = 3 6. A ad A t have the same eigevalues.. Let A ad B be -Matrices. The A B ad B A have the same eigevalues. 3. If x is a eigevector of A correspodig to λ, the x is a eigevector of A k correspodig to eigevalue λ k. 4. If x is a eigevector of regular matrix A correspodig to λ, the x is a eigevector of A correspodig to eigevalue λ. Eigevectors correspodig to eigevalue λ i are all o-vaishig multiples of v i. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 3 / 45 Properties of Eigevalues Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 4 / 45 Eigevalues of Similar Matrices 5. The product of all eigevalues λ i of a matrix A is equal to the determiat of A: det(a) = λ i i= This implies: A is regular if ad oly if all its eigevalues are o-zero. 6. The sum of all eigevalues λ i of a matrix A is equal to the sum of the diagoal elemets of A (called the trace of A). tr(a) = i= a ii = λ i i= Let U be the trasformatio matrix ad C = U A U. If x is a eigevector of A correspodig to eigevalue λ, the U x is a eigevector of C correspodig to λ: C (U x) = (U AU)U x = U Ax = U λx = λ (U x) Similar matrices A ad C have the same eigevalues ad (if we cosider chage of basis) the same eigevectors. We wat to choose a basis such that the matrix that represets the give liear map becomes as simple as possible. The simplest matrices are diagoal matrices. Ca we (always) fid a represetatio by meas of a diagoal matrix? Ufortuately ot i the geeral case. But... Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 5 / 45 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 6 / 45
3 Symmetric Matrix A matrix A is called symmetric, if A t = A. For a symmetric matrix A we fid: All eigevalues are real. Eigevectors u i correspodig to distict eigevalues λ i are orthogoal (i.e., u t i u j = 0 if i = j). There exists a orthoormal basis {u,..., u } (i.e. the vectors u i are ormalized ad mutually orthogoal) that cosists of eigevectors of A, Trasformatio matrix U = (u,..., u ) is the a orthogoal matrix: U t U = I U = U t Diagoalizatio For the i-th uit vector e i we fid Ad thus U t A U e i = U t A u i = U t λ i u i = λ i U t u i = λ i e i λ U t 0 λ A U = D = λ Every symmetric matrix A becomes a diagoal matrix with the eigevalues of A as its etries if we use the orthoormal basis of eigevectors. This procedure is called diagoalizatio of matrix A. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 7 / 45 Example Diagoalizatio We wat to diagoalize A =. Eigevalues λ = ad λ = 3 with respective ormalized eigevectors u = ad u = ( With respect to basis {u, u } matrix A becomes diagoal matrix 0 ) Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 8 / 45 A Geometric Iterpretatio I Fuctio x Ax = x maps the uit circle i R ito a ellipsis. The two semi-axes of the ellipsis are give by λ v ad λ v, resp. v v A v 3v 0 3 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 9 / 45 Quadratic Form Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 0 / 45 Example Quadratic Form Let A be a symmetric matrix. The fuctio q A : R R, x q A (x) = x t A x is called a quadratic form. 0 0 Let A = 0 0. The t x 0 0 q A (x) = x = + x + 3 I geeral we fid for matrix A = (a ij ): q A (x) = = q A (x) = x x i= j= a ij x i x j t 3 3 x t x + x x = + x 4x Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues / 45 Defiiteess Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues / 45 Defiiteess A quadratic form q A is called positive defiite, if for all x = 0, q A (x) > 0. positive semidefiite, if for all x, q A (x) 0. egative defiite, if for all x = 0, q A (x) < 0. egative semidefiite, if for all x, q A (x) 0. idefiite i all other cases. A matrix A is called positive (egative) defiite (semidefiite), if the correspodig quadratic form is positive (egative) defiite (semidefiite). Every symmetric matrix is diagoalizable. Let {u,..., u } be the orthoormal basis of eigevectors of A. The for every x: x = i= c i (x)u i = Uc(x) U = (u,..., u ) is the trasformatio matrix for the orthoormal basis, c the correspodig coefficiet vector. So if D is the diagoal matrix of eigevalues λ i of A we fid q A (x) = x t A x = (Uc) t A Uc = c t U t AU c = c t D c ad thus q A (x) = i= c i λ i Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 3 / 45 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 4 / 45
4 Defiiteess ad Eigevalues Equatio q A (x) = i= c i λ i immediately implies: Let λ i be the eigevalues of symmetric matrix A. The A (the quadratic form q A ) is positive defiite, if all λ i > 0. positive semidefiite, if all λ i 0. egative defiite, if all λ i < 0. egative semidefiite, if all λ i 0. idefiite, if there exist λ i > 0 ad λ j < 0. Example Defiiteess ad Eigevalues The eigevalues of are λ = 6 ad λ =. 5 Thus the matrix is positive defiite. 5 4 The eigevalues of are 4 5 λ = 0, λ = 3, ad λ 3 = 9. The matrix is positive semidefiite The eigevalues of are 4 4 λ = 6, λ = 6 ad λ 3 =. Thus the matrix is idefiite. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 5 / 45 Leadig Priciple Miors Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 6 / 45 Leadig Priciple Miors ad Defiiteess The defiiteess of a matrix ca also be determied by meas of miors. Let A = (a ij ) be a symmetric matrix. The the determiat of submatrix a... a k A k =..... a k... a kk is called the k-th leadig priciple mior of A. A symmetric Matrix A is positive defiite, if ad oly if all A k > 0. egative defiite, if ad oly if ( ) k A k > 0 for all k. idefiite, if A = 0 ad oe of the two cases is holds. ( ) k A k > 0 meas that A, A 3, A 5,... < 0, ad A, A 4, A 6,... > 0. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 7 / 45 Example Leadig Priciple Miors Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 8 / 45 Example Leadig Priciple Miors Defiiteess of matrix 0 A = 3 0 A = det(a ) = a = > 0 a a A = a a = 3 = 5 > 0 Defiiteess of matrix A = 3 3 A = det(a ) = a = > 0 a a A = a a = = > 0 A ad q A are positive defiite. 0 A 3 = A = 3 = 8 > 0 0 A ad q A are idefiite. A 3 = A = 3 = 8 < 0 3 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 9 / 45 Priciple Miors Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 30 / 45 Priciple Miors ad Semidefiiteess Ufortuately the coditio for semidefiiteess is more tedious. Let A = (a ij ) be a symmetric matrix. The the determiat of submatrix a i,i... a i,i k A i,...,i k =..... a ik,i... a ik,i k is called a priciple mior of order k of A. i <... < i k. A symmetric matrix A is positive semidefiite, if ad oly if all A i,...,i k 0. egative semidefiite, if ad oly if ( ) k A i,...,i k 0 for all k. idefiite i all other cases. ( ) k A i,...,i k 0 meas that A i,...,i k 0, if k is eve, ad A i,...,i k 0, if k is odd. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 3 / 45 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 3 / 45
5 Example Priciple Miors Example Priciple Miors Defiiteess of matrix 5 4 A = 4 5 The matrix is positive semidefiite. (But ot positive defiite!) priciple miors of order : A = 5 0 A = 0 A 3 = 5 0 priciple miors of order : 5 A, = = A,3 = 4 5 = 9 0 A,3 = 5 = 9 0 Defiiteess of matrix 5 4 A = 4 5 The matrix is egative semidefiite. (But ot egative defiite!) priciple miors of order : A = 5 0 A = 0 A 3 = 5 0 priciple miors of order : 5 A, = = A,3 = 4 5 = 9 0 A,3 = 5 = 9 0 priciple miors of order 3: A,,3 = A = 0 0 priciple miors of order 3: A,,3 = A = 0 0 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 33 / 45 Leadig Priciple Miors ad Semidefiiteess Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 34 / 45 Recipe for Semidefiiteess Obviously every positive defiite matrix is also positive semidefiite (but ot ecessarily the other way roud). Matrix 0 A = 3 0 is positive defiite as all leadig priciple miors are positive (see above). Therefore A is also positive semidefiite. I this case there is o eed to compute the o-leadig priciple miors. Recipe for fidig semidefiiteess of matrix A:. Compute all leadig priciple miors: If the coditio for positive defiiteess holds, the A is positive defiite ad thus positive semidefiite. Else if the coditio for egative defiiteess holds, the A is egative defiite ad thus egative semidefiite. Else if det(a) = 0, the A is idefiite.. Else also compute all o-leadig priciple miors: If the coditio for positive semidefiiteess holds, the A is positive semidefiite. Else if the coditio for egative semidefiiteess holds, the A is egative semidefiite. Else A is idefiite. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 35 / 45 Ellipse Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 36 / 45 A Geometric Iterpretatio II Equatio a + by =, a, b > 0 describes a ellipse i caoical form. Term a + by is a quadratic form with matrix a 0 A = 0 b / b / a It has eigevalues ad ormalized eigevectors λ = a with v = e ad λ = b with v = e. The semi-axes have legth a ad b, resp. λ v λ v These eigevectors coicide with the semi-axes of the ellipse. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 37 / 45 A Geometric Iterpretatio II Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 38 / 45 A Geometric Iterpretatio II Now let A be a symmetric matrix with positive eigevalues. Equatio x t Ax = describes a ellipse where the semi-axes (priciple axes) coicide coicides with its ormalized eigevectors as see below. λ v λ v By a chage of basis from {e, e } to {v, v } by meas of trasformatio U = (v, v ) this ellipse is rotated ito caoical form. λ v λ v U t λ e λ e (That is, we have diagoalize matrix A.) Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 39 / 45 Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 40 / 45
6 A Applicatio i Statistics Suppose we have observatios of k metric attributes X,..., X k which we combie ito a vector: x i = (x i,..., x ik ) R k A Applicatio i Statistics A chage of basis by meas of a orthogoal matrix does ot chage TSS. However, it chages the cotributios of each of the compoets. The arithmetic mea the is (as for uivariate data) x = i= x i = (x,..., x k ) The total sum of squares is a measure for the statistical dispersio TSS = i= x i x = ad ca be computed compoet-wise. ( k ) x ij x j = j= i= k j= TSS j Ca we fid a basis such that a few compoets cotribute much more to the TSS tha the remaiig oes? Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 4 / 45 Priciple Compoet Aalysis (PCA) Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 4 / 45 Priciple Compoet Aalysis (PCA) Assumptios: The data are approximately multiormal distributed. Procedure:. Compute the covariace matrix Σ.. Compute all eigevalues ad ormalized eigevectors of Σ. 3. Sort eigevalues such that λ λ... λ k. 4. Use correspodig eigevectors v,..., v k as ew basis. 5. The cotributio to TSS of the first m compoets i this basis is m j= λ j m j= TSSj k j= TSSj k j= λ. j By meas of PCA it is possible to reduce the umber of dimesios without reducig the TSS substatially. Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 43 / 45 Summary Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 44 / 45 Eigevalues ad eigevectors Characteristic polyomial Eigespace Properties of eigevalues Symmetric Matrices ad Diagoalizatio Quadratic forms Defiitess Priciple miors Priciple compoet aalysis Josef Leydold Mathematical Methods WS 08/9 6 Eigevalues 45 / 45
Chapter 6. Eigenvalues. Josef Leydold Mathematical Methods WS 2018/19 6 Eigenvalues 1 / 45
Chapter 6 Eigenvalues Josef Leydold Mathematical Methods WS 2018/19 6 Eigenvalues 1 / 45 Closed Leontief Model In a closed Leontief input-output-model consumption and production coincide, i.e. V x = x
More information1 Last time: similar and diagonalizable matrices
Last time: similar ad diagoalizable matrices Let be a positive iteger Suppose A is a matrix, v R, ad λ R Recall that v a eigevector for A with eigevalue λ if v ad Av λv, or equivaletly if v is a ozero
More informationSymmetric Matrices and Quadratic Forms
7 Symmetric Matrices ad Quadratic Forms 7.1 DIAGONALIZAION OF SYMMERIC MARICES SYMMERIC MARIX A symmetric matrix is a matrix A such that. A = A Such a matrix is ecessarily square. Its mai diagoal etries
More informationApply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
More informationLecture 8: October 20, Applications of SVD: least squares approximation
Mathematical Toolkit Autum 2016 Lecturer: Madhur Tulsiai Lecture 8: October 20, 2016 1 Applicatios of SVD: least squares approximatio We discuss aother applicatio of sigular value decompositio (SVD) of
More information5.1. The Rayleigh s quotient. Definition 49. Let A = A be a self-adjoint matrix. quotient is the function. R(x) = x,ax, for x = 0.
40 RODICA D. COSTIN 5. The Rayleigh s priciple ad the i priciple for the eigevalues of a self-adjoit matrix Eigevalues of self-adjoit matrices are easy to calculate. This sectio shows how this is doe usig
More informationTopics in Eigen-analysis
Topics i Eige-aalysis Li Zajiag 28 July 2014 Cotets 1 Termiology... 2 2 Some Basic Properties ad Results... 2 3 Eige-properties of Hermitia Matrices... 5 3.1 Basic Theorems... 5 3.2 Quadratic Forms & Noegative
More informationThe Method of Least Squares. To understand least squares fitting of data.
The Method of Least Squares KEY WORDS Curve fittig, least square GOAL To uderstad least squares fittig of data To uderstad the least squares solutio of icosistet systems of liear equatios 1 Motivatio Curve
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More informationChimica Inorganica 3
himica Iorgaica Irreducible Represetatios ad haracter Tables Rather tha usig geometrical operatios, it is ofte much more coveiet to employ a ew set of group elemets which are matrices ad to make the rule
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture & 3: Pricipal Compoet Aalysis The text i black outlies high level ideas. The text i blue provides simple mathematical details to derive or get to the algorithm
More informationRandom Matrices with Blocks of Intermediate Scale Strongly Correlated Band Matrices
Radom Matrices with Blocks of Itermediate Scale Strogly Correlated Bad Matrices Jiayi Tog Advisor: Dr. Todd Kemp May 30, 07 Departmet of Mathematics Uiversity of Califoria, Sa Diego Cotets Itroductio Notatio
More informationPROBLEM SET I (Suggested Solutions)
Eco3-Fall3 PROBLE SET I (Suggested Solutios). a) Cosider the followig: x x = x The quadratic form = T x x is the required oe i matrix form. Similarly, for the followig parts: x 5 b) x = = x c) x x x x
More informationLINEAR ALGEBRA. Paul Dawkins
LINEAR ALGEBRA Paul Dawkis Table of Cotets Preface... ii Outlie... iii Systems of Equatios ad Matrices... Itroductio... Systems of Equatios... Solvig Systems of Equatios... 5 Matrices... 7 Matrix Arithmetic
More informationNBHM QUESTION 2007 Section 1 : Algebra Q1. Let G be a group of order n. Which of the following conditions imply that G is abelian?
NBHM QUESTION 7 NBHM QUESTION 7 NBHM QUESTION 7 Sectio : Algebra Q Let G be a group of order Which of the followig coditios imply that G is abelia? 5 36 Q Which of the followig subgroups are ecesarily
More informationWhere do eigenvalues/eigenvectors/eigenfunctions come from, and why are they important anyway?
Where do eigevalues/eigevectors/eigeuctios come rom, ad why are they importat ayway? I. Bacgroud (rom Ordiary Dieretial Equatios} Cosider the simplest example o a harmoic oscillator (thi o a vibratig strig)
More informationSingular value decomposition. Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaine
Lecture 11 Sigular value decompositio Mathématiques appliquées (MATH0504-1) B. Dewals, Ch. Geuzaie V1.2 07/12/2018 1 Sigular value decompositio (SVD) at a glace Motivatio: the image of the uit sphere S
More informationInverse Matrix. A meaning that matrix B is an inverse of matrix A.
Iverse Matrix Two square matrices A ad B of dimesios are called iverses to oe aother if the followig holds, AB BA I (11) The otio is dual but we ofte write 1 B A meaig that matrix B is a iverse of matrix
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationTMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods
TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A
More informationMon Apr Second derivative test, and maybe another conic diagonalization example. Announcements: Warm-up Exercise:
Math 2270-004 Week 15 otes We will ot ecessarily iish the material rom a give day's otes o that day We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie
More informationCov(aX, cy ) Var(X) Var(Y ) It is completely invariant to affine transformations: for any a, b, c, d R, ρ(ax + b, cy + d) = a.s. X i. as n.
CS 189 Itroductio to Machie Learig Sprig 218 Note 11 1 Caoical Correlatio Aalysis The Pearso Correlatio Coefficiet ρ(x, Y ) is a way to measure how liearly related (i other words, how well a liear model
More informationMATH10212 Linear Algebra B Proof Problems
MATH22 Liear Algebra Proof Problems 5 Jue 26 Each problem requests a proof of a simple statemet Problems placed lower i the list may use the results of previous oes Matrices ermiats If a b R the matrix
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math S-b Lecture # Notes This wee is all about determiats We ll discuss how to defie them, how to calculate them, lear the allimportat property ow as multiliearity, ad show that a square matrix A is ivertible
More informationOn Nonsingularity of Saddle Point Matrices. with Vectors of Ones
Iteratioal Joural of Algebra, Vol. 2, 2008, o. 4, 197-204 O Nosigularity of Saddle Poit Matrices with Vectors of Oes Tadeusz Ostrowski Istitute of Maagemet The State Vocatioal Uiversity -400 Gorzów, Polad
More informationLemma Let f(x) K[x] be a separable polynomial of degree n. Then the Galois group is a subgroup of S n, the permutations of the roots.
15 Cubics, Quartics ad Polygos It is iterestig to chase through the argumets of 14 ad see how this affects solvig polyomial equatios i specific examples We make a global assumptio that the characteristic
More informationLECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK)
LECTURE 8: ORTHOGONALITY (CHAPTER 5 IN THE BOOK) Everythig marked by is ot required by the course syllabus I this lecture, all vector spaces is over the real umber R. All vectors i R is viewed as a colum
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math E-2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A
More informationLinear regression. Daniel Hsu (COMS 4771) (y i x T i β)2 2πσ. 2 2σ 2. 1 n. (x T i β y i ) 2. 1 ˆβ arg min. β R n d
Liear regressio Daiel Hsu (COMS 477) Maximum likelihood estimatio Oe of the simplest liear regressio models is the followig: (X, Y ),..., (X, Y ), (X, Y ) are iid radom pairs takig values i R d R, ad Y
More information4 The Sperner property.
4 The Sperer property. I this sectio we cosider a surprisig applicatio of certai adjacecy matrices to some problems i extremal set theory. A importat role will also be played by fiite groups. I geeral,
More informationCOLLIN COUNTY COMMUNITY COLLEGE COURSE SYLLABUS CREDIT HOURS: 3 LECTURE HOURS: 3 LAB HOURS: 0
COLLIN COUNTY COMMUNITY COLLEGE COURSE SYLLABUS Revised Fall 2017 COURSE NUMBER: MATH 2318 COURSE TITLE: Liear Algebra CREDIT HOURS: 3 LECTURE HOURS: 3 LAB HOURS: 0 ASSESSMENTS: Noe PREREQUISITE: MATH
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationOutline. Linear regression. Regularization functions. Polynomial curve fitting. Stochastic gradient descent for regression. MLE for regression
REGRESSION 1 Outlie Liear regressio Regularizatio fuctios Polyomial curve fittig Stochastic gradiet descet for regressio MLE for regressio Step-wise forward regressio Regressio methods Statistical techiques
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationMachine Learning for Data Science (CS 4786)
Machie Learig for Data Sciece CS 4786) Lecture 9: Pricipal Compoet Aalysis The text i black outlies mai ideas to retai from the lecture. The text i blue give a deeper uderstadig of how we derive or get
More informationThe Jordan Normal Form: A General Approach to Solving Homogeneous Linear Systems. Mike Raugh. March 20, 2005
The Jorda Normal Form: A Geeral Approach to Solvig Homogeeous Liear Sstems Mike Raugh March 2, 25 What are we doig here? I this ote, we describe the Jorda ormal form of a matrix ad show how it ma be used
More informationChapter 12 EM algorithms The Expectation-Maximization (EM) algorithm is a maximum likelihood method for models that have hidden variables eg. Gaussian
Chapter 2 EM algorithms The Expectatio-Maximizatio (EM) algorithm is a maximum likelihood method for models that have hidde variables eg. Gaussia Mixture Models (GMMs), Liear Dyamic Systems (LDSs) ad Hidde
More informationThe Discrete Fourier Transform
The Discrete Fourier Trasform Complex Fourier Series Represetatio Recall that a Fourier series has the form a 0 + a k cos(kt) + k=1 b k si(kt) This represetatio seems a bit awkward, sice it ivolves two
More informationBertrand s Postulate
Bertrad s Postulate Lola Thompso Ross Program July 3, 2009 Lola Thompso (Ross Program Bertrad s Postulate July 3, 2009 1 / 33 Bertrad s Postulate I ve said it oce ad I ll say it agai: There s always a
More informationChapter Unary Matrix Operations
Chapter 04.04 Uary atrix Operatios After readig this chapter, you should be able to:. kow what uary operatios meas, 2. fid the traspose of a square matrix ad it s relatioship to symmetric matrices,. fid
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationCHAPTER 5. Theory and Solution Using Matrix Techniques
A SERIES OF CLASS NOTES FOR 2005-2006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 3 A COLLECTION OF HANDOUTS ON SYSTEMS OF ORDINARY DIFFERENTIAL
More informationState Space Representation
Optimal Cotrol, Guidace ad Estimatio Lecture 2 Overview of SS Approach ad Matrix heory Prof. Radhakat Padhi Dept. of Aerospace Egieerig Idia Istitute of Sciece - Bagalore State Space Represetatio Prof.
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 9 Multicolliearity Dr Shalabh Departmet of Mathematics ad Statistics Idia Istitute of Techology Kapur Multicolliearity diagostics A importat questio that
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig
More informationOrthogonal transformations
Orthogoal trasformatios October 12, 2014 1 Defiig property The squared legth of a vector is give by takig the dot product of a vector with itself, v 2 v v g ij v i v j A orthogoal trasformatio is a liear
More informationSession 5. (1) Principal component analysis and Karhunen-Loève transformation
200 Autum semester Patter Iformatio Processig Topic 2 Image compressio by orthogoal trasformatio Sessio 5 () Pricipal compoet aalysis ad Karhue-Loève trasformatio Topic 2 of this course explais the image
More information17. Joint distributions of extreme order statistics Lehmann 5.1; Ferguson 15
17. Joit distributios of extreme order statistics Lehma 5.1; Ferguso 15 I Example 10., we derived the asymptotic distributio of the maximum from a radom sample from a uiform distributio. We did this usig
More informationAN INTRODUCTION TO SPECTRAL GRAPH THEORY
AN INTRODUCTION TO SPECTRAL GRAPH THEORY JIAQI JIANG Abstract. Spectral graph theory is the study of properties of the Laplacia matrix or adjacecy matrix associated with a graph. I this paper, we focus
More informationdenote the set of all polynomials of the form p=ax 2 +bx+c. For example, . Given any two polynomials p= ax 2 +bx+c and q= a'x 2 +b'x+c',
Chapter Geeral Vector Spaces Real Vector Spaces Example () Let u ad v be vectors i R ad k a scalar ( a real umber), the we ca defie additio: u+v, scalar multiplicatio: ku, kv () Let P deote the set of
More informationCHAPTER 3. GOE and GUE
CHAPTER 3 GOE ad GUE We quicly recall that a GUE matrix ca be defied i the followig three equivalet ways. We leave it to the reader to mae the three aalogous statemets for GOE. I the previous chapters,
More information(VII.A) Review of Orthogonality
VII.A Review of Orthogoality At the begiig of our study of liear trasformatios i we briefly discussed projectios, rotatios ad projectios. I III.A, projectios were treated i the abstract ad without regard
More informationStochastic Matrices in a Finite Field
Stochastic Matrices i a Fiite Field Abstract: I this project we will explore the properties of stochastic matrices i both the real ad the fiite fields. We first explore what properties 2 2 stochastic matrices
More informationBrief Review of Functions of Several Variables
Brief Review of Fuctios of Several Variables Differetiatio Differetiatio Recall, a fuctio f : R R is differetiable at x R if ( ) ( ) lim f x f x 0 exists df ( x) Whe this limit exists we call it or f(
More informationApplication of Jordan Canonical Form
CHAPTER 6 Applicatio of Jorda Caoical Form Notatios R is the set of real umbers C is the set of complex umbers Q is the set of ratioal umbers Z is the set of itegers N is the set of o-egative itegers Z
More informationLinear Classifiers III
Uiversität Potsdam Istitut für Iformatik Lehrstuhl Maschielles Lere Liear Classifiers III Blaie Nelso, Tobias Scheffer Cotets Classificatio Problem Bayesia Classifier Decisio Liear Classifiers, MAP Models
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationCHAPTER I: Vector Spaces
CHAPTER I: Vector Spaces Sectio 1: Itroductio ad Examples This first chapter is largely a review of topics you probably saw i your liear algebra course. So why cover it? (1) Not everyoe remembers everythig
More informationMath E-21b Spring 2018 Homework #2
Math E- Sprig 08 Homework # Prolems due Thursday, Feruary 8: Sectio : y = + 7 8 Fid the iverse of the liear trasformatio [That is, solve for, i terms of y, y ] y = + 0 Cosider the circular face i the accompayig
More informationAfter the completion of this section the student should recall
Chapter III Liear Algebra September 6, 7 6 CHAPTER III LINEAR ALGEBRA Objectives: After the completio of this sectio the studet should recall - the cocept of vector spaces - the operatios with vectors
More informationFilter banks. Separately, the lowpass and highpass filters are not invertible. removes the highest frequency 1/ 2and
Filter bas Separately, the lowpass ad highpass filters are ot ivertible T removes the highest frequecy / ad removes the lowest frequecy Together these filters separate the sigal ito low-frequecy ad high-frequecy
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More information6 Integers Modulo n. integer k can be written as k = qn + r, with q,r, 0 r b. So any integer.
6 Itegers Modulo I Example 2.3(e), we have defied the cogruece of two itegers a,b with respect to a modulus. Let us recall that a b (mod ) meas a b. We have proved that cogruece is a equivalece relatio
More informationMath 220B Final Exam Solutions March 18, 2002
Math 0B Fial Exam Solutios March 18, 00 1. (1 poits) (a) (6 poits) Fid the Gree s fuctio for the tilted half-plae {(x 1, x ) R : x 1 + x > 0}. For x (x 1, x ), y (y 1, y ), express your Gree s fuctio G(x,
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationMon Feb matrix inverses. Announcements: Warm-up Exercise:
Math 225-4 Week 6 otes We will ot ecessarily fiish the material from a give day's otes o that day We may also add or subtract some material as the week progresses, but these otes represet a i-depth outlie
More informationUnit 5. Hypersurfaces
Uit 5. Hyersurfaces ================================================================= -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
More informationPAijpam.eu ON DERIVATION OF RATIONAL SOLUTIONS OF BABBAGE S FUNCTIONAL EQUATION
Iteratioal Joural of Pure ad Applied Mathematics Volume 94 No. 204, 9-20 ISSN: 3-8080 (prited versio); ISSN: 34-3395 (o-lie versio) url: http://www.ijpam.eu doi: http://dx.doi.org/0.2732/ijpam.v94i.2 PAijpam.eu
More informationApplications in Linear Algebra and Uses of Technology
1 TI-89: Let A 1 4 5 6 7 8 10 Applicatios i Liear Algebra ad Uses of Techology,adB 4 1 1 4 type i: [1,,;4,5,6;7,8,10] press: STO type i: A type i: [4,-1;-1,4] press: STO (1) Row Echelo Form: MATH/matrix
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More information5.1 Review of Singular Value Decomposition (SVD)
MGMT 69000: Topics i High-dimesioal Data Aalysis Falll 06 Lecture 5: Spectral Clusterig: Overview (cotd) ad Aalysis Lecturer: Jiamig Xu Scribe: Adarsh Barik, Taotao He, September 3, 06 Outlie Review of
More informationMath 5311 Problem Set #5 Solutions
Math 5311 Problem Set #5 Solutios March 9, 009 Problem 1 O&S 11.1.3 Part (a) Solve with boudary coditios u = 1 0 x < L/ 1 L/ < x L u (0) = u (L) = 0. Let s refer to [0, L/) as regio 1 ad (L/, L] as regio.
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationIn algebra one spends much time finding common denominators and thus simplifying rational expressions. For example:
74 The Method of Partial Fractios I algebra oe speds much time fidig commo deomiators ad thus simplifyig ratioal epressios For eample: + + + 6 5 + = + = = + + + + + ( )( ) 5 It may the seem odd to be watig
More informationLecture 3: Divide and Conquer: Fast Fourier Transform
Lecture 3: Divide ad Coquer: Fast Fourier Trasform Polyomial Operatios vs. Represetatios Divide ad Coquer Algorithm Collapsig Samples / Roots of Uity FFT, IFFT, ad Polyomial Multiplicatio Polyomial operatios
More informationM A T H F A L L CORRECTION. Algebra I 1 4 / 1 0 / U N I V E R S I T Y O F T O R O N T O
M A T H 2 4 0 F A L L 2 0 1 4 HOMEWORK ASSIGNMENT #4 CORRECTION Algebra I 1 4 / 1 0 / 2 0 1 4 U N I V E R S I T Y O F T O R O N T O P r o f e s s o r : D r o r B a r - N a t a Correctio Homework Assigmet
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationMaths /2014. CCP Maths 2. Reduction, projector,endomorphism of rank 1... Hadamard s inequality and some applications. Solution.
CCP Maths 2 Reductio, projector,edomorphism of rak 1... Hadamard s iequality ad some applicatios Solutio Exercise 1. 1 A is a symmetric matrix so diagoalizable. 2 Diagoalizatio of A : A characteristic
More information, then cv V. Differential Equations Elements of Lineaer Algebra Name: Consider the differential equation. and y2 cos( kx)
Cosider the differetial equatio y '' k y 0 has particular solutios y1 si( kx) ad y cos( kx) I geeral, ay liear combiatio of y1 ad y, cy 1 1 cy where c1, c is also a solutio to the equatio above The reaso
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More information6. Kalman filter implementation for linear algebraic equations. Karhunen-Loeve decomposition
6. Kalma filter implemetatio for liear algebraic equatios. Karhue-Loeve decompositio 6.1. Solvable liear algebraic systems. Probabilistic iterpretatio. Let A be a quadratic matrix (ot obligatory osigular.
More informationLinear Transformations
Liear rasformatios 6. Itroductio to Liear rasformatios 6. he Kerel ad Rage of a Liear rasformatio 6. Matrices for Liear rasformatios 6.4 rasitio Matrices ad Similarity 6.5 Applicatios of Liear rasformatios
More informationCALCULUS BASIC SUMMER REVIEW
CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=
More informationWhy learn matrix algebra? Vectors & Matrices with statistical applications. Brief history of linear algebra
R Vectors & Matrices with statistical applicatios x RXX RXY y RYX RYY Why lear matrix algebra? Simple way to express liear combiatios of variables ad geeral solutios of equatios. Liear statistical models
More informationPAijpam.eu ON TENSOR PRODUCT DECOMPOSITION
Iteratioal Joural of Pure ad Applied Mathematics Volume 103 No 3 2015, 537-545 ISSN: 1311-8080 (prited versio); ISSN: 1314-3395 (o-lie versio) url: http://wwwijpameu doi: http://dxdoiorg/1012732/ijpamv103i314
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 5-4 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig facts,
More informationSignal Processing in Mechatronics
Sigal Processig i Mechatroics Zhu K.P. AIS, UM. Lecture, Brief itroductio to Sigals ad Systems, Review of Liear Algebra ad Sigal Processig Related Mathematics . Brief Itroductio to Sigals What is sigal
More informationPolynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
More information(3) If you replace row i of A by its sum with a multiple of another row, then the determinant is unchanged! Expand across the i th row:
Math 50-004 Tue Feb 4 Cotiue with sectio 36 Determiats The effective way to compute determiats for larger-sized matrices without lots of zeroes is to ot use the defiitio, but rather to use the followig
More informationsubcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016
subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationMatrix Algebra from a Statistician s Perspective BIOS 524/ Scalar multiple: ka
Matrix Algebra from a Statisticia s Perspective BIOS 524/546. Matrices... Basic Termiology a a A = ( aij ) deotes a m matrix of values. Whe =, this is a am a m colum vector. Whe m= this is a row vector..2.
More information24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS
24 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS Corollary 2.30. Suppose that the semisimple decompositio of the G- module V is V = i S i. The i = χ V,χ i Proof. Sice χ V W = χ V + χ W, we have:
More informationMath 61CM - Solutions to homework 3
Math 6CM - Solutios to homework 3 Cédric De Groote October 2 th, 208 Problem : Let F be a field, m 0 a fixed oegative iteger ad let V = {a 0 + a x + + a m x m a 0,, a m F} be the vector space cosistig
More informationECE 308 Discrete-Time Signals and Systems
ECE 38-5 ECE 38 Discrete-Time Sigals ad Systems Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa ECE 38-5 1 Additio, Multiplicatio, ad Scalig of Sequeces Amplitude Scalig: (A Costat
More informationAH Checklist (Unit 3) AH Checklist (Unit 3) Matrices
AH Checklist (Uit 3) AH Checklist (Uit 3) Matrices Skill Achieved? Kow that a matrix is a rectagular array of umbers (aka etries or elemets) i paretheses, each etry beig i a particular row ad colum Kow
More informationAssignment 2 Solutions SOLUTION. ϕ 1 Â = 3 ϕ 1 4i ϕ 2. The other case can be dealt with in a similar way. { ϕ 2 Â} χ = { 4i ϕ 1 3 ϕ 2 } χ.
PHYSICS 34 QUANTUM PHYSICS II (25) Assigmet 2 Solutios 1. With respect to a pair of orthoormal vectors ϕ 1 ad ϕ 2 that spa the Hilbert space H of a certai system, the operator  is defied by its actio
More informationMachine Learning Theory Tübingen University, WS 2016/2017 Lecture 11
Machie Learig Theory Tübige Uiversity, WS 06/07 Lecture Tolstikhi Ilya Abstract We will itroduce the otio of reproducig kerels ad associated Reproducig Kerel Hilbert Spaces (RKHS). We will cosider couple
More information