The Group Inverse of extended Symmetric and Periodic Jacobi Matrices

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1 GRUPO MAPTHE Preprint 206 February 3, 206. The Group Inverse of extended Symmetric and Periodic Jacobi Matrices S. Gago Abstract. In this work, we explicitly compute the group inverse of symmetric and periodic Jacobi matrices with constant elements that have been extended by adding a row and a column conveniently defined. For this purpose, we interpret such matrices as the combinatorial Laplacian of a non complete wheel that has been obtained by adding a vertex to a cycle and some edges conveniently chosen. The obtained group inverse is an incomplete block matrix with a block Toeplitz structure. In addition, we obtain the effective resistances and the Kirchhoff index of non complete wheels.. Introduction and notation The invertibility of nonsingular tridiagonal or block tridiagonal matrices has been studied in recent years; see 3 for a review on the subject. Moreover, explicit inverses are known in some cases, for instance when the tridiagonal matrix is symmetric with constant diagonals and subject to some restrictions. In 9, da Fonseca and Petronilho obtained explicit inverses of tridiagonal 2 Toeplitz and 3 Toeplitz matrices which generalize some well-known results concerning the inverse of a tridiagonal Toeplitz matrix. Recently, M.J. Jiménez in her PhD Tesis gave an explicit formula for the inverse of p, r Toeplitz Jacobi matrices tridiagonal matrices whose diagonals are formed by quasi periodic sequences, see,2. The techniques used in the mentioned results are mainly based on the theory of orthogonal polynomials. For the singular case, there is also a big amount of work, for instance in 4 the authors carried out an exhaustive analysis of the generalized inverses of singular irreducible symmetric M matrices. The key idea of the approach was to identify any symmetric M matrix with a positive semi definite Schrödinger operator on a connected network whose conductances are given by the off diagonal elements of the M matrix. Moreover, explicit expressions for the group inverses in the cases of tridiagonal matrices and circulant matrices were obtained in 6. In this work, we present a new formula for the group inverse of an extended symmetric and periodic Jacobi matrix with constant elements. By extended we mean that a row and a column have been added to the matrix. The idea is to see the extended symmetric and periodic Jacobi matrix as the combinatorial Laplacian of a non complete wheel and hence to obtain the group inverse, by using some previous results obtained in 7. A non complete wheel is a wheel where the central vertex is connected to a few vertices of the cycle. This kind of networks has many applications in Computer Science as the central vertex is called a hub see 2. Therefore, the non complete wheel can be seen as a cycle with an added vertex and some new edges and hence, the result of 7 can be applied. Finally, we use the formula for the group inverse to give the effective resistances and the Kirchhoff Index a non complete wheel Mathematics Subject Classification: Keywords: Symmetric and Circulant Matrices, Inverses, Chebyshev polynomials.

2 2 S. Gago In the following, the triple Γ = V, E, c denotes a finite network, that is, a finite graph without loops nor multiple edges, with vertex set V = {,..., n} and edge set E, where each edge e ij = {i, j} has associated a conductance c ij > 0. The standard inner product on R n is denoted by,, thus, if u, v R n, then u, v = n k= u kv k. For any i =,..., n we denote by e i the i th vector of the standard basis of R n, by j the all ones vector of dimension n and by J the all ones matrix. A matrix is called a block Toeplitz matrix iff is a block matrix, which contains blocks that are repeated in each descending diagonal from left to right, as a Toeplitz matrix in which each descending diagonal from left to right is constant. A matrix A of order n = md + p, 0 < p < d, is called incomplete block matrix if it is partitioned from the top left hand corner using d d submatrices as far as possible. Thus A A m A m A =, A m A mm A mm+ A m+ A m+m A m+m+ where A ij, i, j =, 2,..., m is an d d matrix, A im+ and A m+j, i, j =, 2,..., m are d p matrices and A m+m+ is an p p matrix. The combinatorial Laplacian or simply the Laplacian of the network Γ is the matrix L = L ij, where L ii = n k= c ik, and L ij = c ij when i j. It is well known that the Laplacian is singular, symmetric and positive semi definite. Moreover Lu = 0 iff u is proportional to vector j. The group inverse of the Laplacian, L #, is known as the Green matrix of the network G. For every pair of vertices {i, j} we define the dipole between them as the vector π ij = e i e j. Observe that π ii = 0 and π ij = π ji. The effective resistance between two vertices {i, j} of a network Γ, is defined as Ri, j = Gπ ij, π ij = G ii + G jj 2G ij, and the total resistance of the network or Kirchhoff index is defined as KΓ = n Ri, j = n G ii. 2 i,j V Given m >, d, we consider a cycle C n on n = md vertices with constant conductances c = ci, i + > 0 for any i =,..., n. A non complete wheel n, m-w, is a network obtained from C n by adding a new vertex n + to m of the vertices of the cycle placed at the same distance, d, with new conductances a = cn +, + di, for any i =,..., m. It is known that the Laplacian matrix of the cycle C n, is a circulant matrix L = circ2c, c, 0,..., 0, c and its group inverse is G ij = n 2 6 i j n i j, i, j =,..., n, see 4. i= 2. The group inverse of non complete wheels In this section we give an explicit expression for the group inverse of extended symmetric and periodic Jacobi matrices. Not surprisingly, the expression is an incomplete block Toeplitz matrix whose coefficients involve Chebyshev polinomials. In order to obtain the claimed expression we consider the Laplacian matrix of the non complete wheel network n, m-w in terms of the Laplacian of the base cycle. Then, we obtain the matrix of order n + given by L + D s L = s, α

3 The inverse matrix of some circulant matrices 3 where D is a diagonal matrix whose non null elements, a, are placed at the + di elements of the diagonal, i =,..., m, s = a m e +di and α = ma. i= Theorem. The group inverse of L is where L # = L L 2 L 2 L 22 L 22 = 2cdn + and2 2acn + 2, L d a 2 = an + 2 2cd d2 + n + 2 n j m, with n i = a n6 + d + 5m + 6mii d 2, for any i =,..., d, and where L is the block Toeplitz matrix N N 2... N m. N L = m N N2 such that any submatrix N k, k =,..., m, has entries N k i h+ = + T m q N 2... N m N i kd h n i kd h an n h i + V k q V m k q c i h d nd U k 2 q + U m k q 2cd a V m q d 2 U m q d 2 k 2 + n + with i =,..., d h = 0,..., d and q = ad 2c +., + dh + nh nd 2 kd 3 d + 2 h n i + kd n 2 h n d 2 ad i 2 di h cd + dh d 6 6an + 2, Proof. In order to obtain L #, we use 7, Corollary which reads: L # n 2 CMC + α n J = 2 n α n j CMs α 2 + n 2, α n j s MC α + s Ms where C = αn + In + js, n and M = G GΠI + Π GΠ Π G, with Π the matrix whose columns are the dipoles a m π +i d +j d, for any i < j m.

4 4 S. Gago So, to obtain the result we basically need to compute matrix M = G GΠI + Π GΠ Π G. To do this we need to calculate the different involved matrices and their products in different steps. For the sake of readability, we develop the proofs in Section 4, throughout some technical lemmas. i In order to obtain B = I + Π GΠ, we use a reduced form Π GΠ = Π R G R Π R, ii where G R is the m m submatrix of G n2 J whose rows and columns are placed at + k d, for any k =,..., m, and Π R is the submatrix of Π obtained only by the m rows placed at + k d, for any k =,..., m Now we apply Woodbury s formula, see 5, to get B = I + Π R G R Π R = I Π R G R + Π R Π R Π R. Thus, M becomes where M R = G R + Π R Π R. M = G GΠI Π R G R + Π R Π R Π R Π G = G GΠΠ ΠΠ R G R + Π R Π R Π R Π G = = G GΠΠ ΠΠ R M R Π R Π G, Observe that both G R and Π R Π R are, respectively, the circulant matrices G R = d circ0, m,..., jm j,..., m, 2cm Π R Π R = a m mi J = a circ m,,...,. m Thus, for computing G R and G R + Π R Π R, we use Lemma. iii Now, from 6, Theorem 3.5, we obtain M R in Lemma 2. iv We next consider F = ΠΠ ΠΠ R M R Π R Π. Notice that ΠΠ is a matrix which has the circulant matrix a mcirc m,,..., as a submatrix at the rows and columns placed at + k d, for any k =,..., m, and 0 otherwise. It turns out that, F is the Toeplitz matrix described in Lemma 3. v Since, M = G GFG, we first compute matrix H = GF, see Lemma 4. vi Next, we compute matrix K = GFG = HG in Lemma 5. vii Again, M is a block Toeplitz matrix whose expression is obtained in Proposition 2. viii Finally, using Lemma 6, the claimed result of Theorem follows. 3. Effective resistances and Kirchhoff index for non complete wheels The Green matrix is a fundamental tool for computing some desired parameters of the network, like the effective resistances of the network or the Kirchhoff index, very useful in electric circuit theory or in organic chemistry, as natural indexes describing important structural properties of circuits or molecules. Therefore, once we have compute the Green matrix, the effective resistances between two vertices of the new network are easily obtained from Formula obtained in 3. Proposition. Given two vertices of Γ, the effective resistances of the network between them are:

5 The inverse matrix of some circulant matrices 5 a if i, j n +, i = k d + h, j = k 2 d + h 2 and k = k 2 k +, where k k 2 n and h, h 2 = 0,..., d, with h < h 2 when k = k 2, then Ri, j = + T m q a c h 2 h V k q V m k q c 2 h h 2 d d c + cn a 2c 2 h2 + h 2 2 h + h 2 d d U m q c U k 2q + U m k q h h 2 k d n h h 2 k d 2 k d + 2 h 2 h k d n 2 h 2 + d 2 k 2 + h 2 + h h 2h 2 d + nh h 2 + d n 2 d 2 6cn + 2, b if i = k d + h n + and j = n +, then d Ri, n + = T m q an V an m q c h2 h d nd + d 2 U m q + h2 h d cn cdn2 + + an 2 d 2 6h 2 6h d ad 2 2acnn + 2 6cn n + 2, 2cd with q = ad 2c +. Moreover, the Kirchoff index of Γ is KΓ = n + an 2c T m q 6c dn + d2 U m q + 2cd a + dn V m q + d dnn a 6c d 2 n. 2c Proof. The proof is straightforward if we observe that for i, j n+, then i = k d+h, j = k 2 d+h 2 for any k k 2 n and h, h 2 = 0,..., d, with h < h 2 when k = k 2, and if k = k 2 k +, then Ri, j = N h+,h + + N h2+,h 2+ 2N k2 k + h+,h 2+, and for i = k d + h n + and j = n +, then Ri, n + = N h+,h + + L 22 2L 2 h+, and simplifying we obtain the claimed result. Similarly, the Kirchoff index of the non complete wheel n, m-w is obtained by simplifying the expression KΓ = n + tracel = n + tracel + n + L 22. Finally, if we consider the case where d =, n = m, we obtain a complete Wheel, and in this case we notice that the Kirchhoff index coincides with the result obtained in 3. Corollary. The Kirchhoff index of a Wheel on n + vertices is KΓ = n + an 2c 2c T n q 6c n + U n q + a + n 3 V n q a nn c 4. Technical lemmas In this section we include the technical lemmas and their proofs. Observe that the first Lemma excludes the case of adding a pendant vertex m =. Besides, we point out that the result of the second Lemma 2 is obtained from the application of Theorem 3.5. from 6.

6 6 S. Gago Lemma. The inverse matrix of G R is the circulant matrix G 2c R = nm 2 circ b 0, b,,...,, b, where b 0 = m3 m 6 and b = m3 m + 2. On the other hand 6 2 G R + Π R Π R = nm 2 circ c 0, c, c 2,..., c 2, c, where c 0 = 2cb 0 + adm m 2, c = 2cb + adm 2, c 2 = 2c + adm 2. Lemma 2. The circulant matrix G R + Π R Π R is invertible iff m > and in this case where M R = G R + Π R Π R = circm,..., m m, Uj 2 q + U m j qd m j = 2cT m q with q = ad 2c +. Lemma 3. The matrix F is a block Toeplitz matrix given by F F 2... F m. F F = m F F2 2c + adm2, j =,..., m, 2acm F 2... F m F where each submatrix F i has all its elements equal 0 except the first one, and with q = ad 2c +. Proof. Observe that F i = f i = aδ i a2 d 2c f = am m f i = a m + a2 m U i 2 q + U m i q, for any i =,..., m, T m q + a2 m m j a 2 m, j= m j a 2 m i, for any i = 2,..., d. Therefore, we first compute Uj 2 q + U m j qd m j = 2c + adm2 2cT j= j= m q 2acm d = U j 2 q + U m j q 2c + adm2 2cT m q 2ac = j= j= d 2c 2cT m q ad T mq 2c + adm2 2ac = d 2c m2.

7 The inverse matrix of some circulant matrices 7 And now we have f = am m and for any i = 2,..., d, f i = a m a2 dm 2 2cm a2 dm 2 2cm a 2 Um qd 2cT m q 2c + adm2 2acm a 2 Ui 2 q + U m i qd 2cT m q 2c + adm2 2acm Lemma 4. The matrix H is a block Toeplitz matrix given by H H 2... H m. H H = m H H2 H 2... H m H = a a2 d 2c = a2 d 2c U m q T m q, U i 2 q + U m i q. T m q where each submatrix H k has all its elements equal 0 except its first column, and for any i =,..., d H i = d n a i V m q du m q, and for any k = 2,..., d, 2c T m q H k i = d n a i V k q V m k q d 2c T m q c ai + cu k 2q + U m k q, with q = ad 2c +. Proof. We point out that G is also a block Toeplitz matrix and thus H is again a block Toeplitz matrix, and hence we need to compute just the first d rows. We first define the following summations: s i, j = j f p, s 2 i, j = p=i j pf p and s 3 i, j = p=i j p 2 f p. p=i In particular, K = s, m = a a 2 T m q 2 T mq = 0, K 2 = s 2, m = a m V m q, 2 T m q K 3 = s 3, m = = 2c d a 2 T mq m 2 V m q 2mU m 2 q mm + 2. And moreover, B = s k +, m = a 2 T m q T m q B 2 = s 2 k +, m = a 2 +U k 2 q + U m k q U m 2 q m +. V m q V k q + V m k q, mv m q + kv m k q V k q

8 8 S. Gago Now observe that when j = + k d and k =,..., m, for i =,..., d we have that H k i = + g i,+l d f k l+ + g i,+l d f m+k l+ g i, f k + g i,+l d f k l+ l= g i,+l d f m+k l+ = A + A 2 + A 3. Firstly we suppose k >. We compute separately A 2 and A 3 as follows A 2 = A 22 = A 23 = k k f k l+ = k p= f p = s, k, k pf p = js, k s 2, k, p= k p 2 f p = k 2 s, k 2ks 2, k + s 3, k, k p= and thus, A 2 = g i,+l d f k l+ = α A 2 + α 2 A 22 + α 3 A 23 = + α 3 s 3, k. n 2 + 6i n + i A 2 6dn + 2i 2A d 2 A 23 = α + α 2 k + α 3 k 2 s, k α 2 + 2α 3 ks 2, k Besides, A 3 = A 32 = A 33 = f m+k l+ = m p=k+ f p = s k +, m, l f m+k l+ = m + ks k +, m s 2 k +, m, l 2 f m+k l+ = m + k 2 s k +, m 2m + ks 2 k +, m + s 3 k +, m, and thus, A 3 = = = g i,+l d f m+k l+ = α A 3 + α 2 A 32 + α 3 A 33 α + α 2 m + k + α 3 m + k 2 s k +, m α 2 + 2α 3 m + ks 2 k +, m + α 3 s 3 k +, m α + α 2 k + α 3 k 2 s k +, m α 2 + 2α 3 ks 2 k +, m + α 3 s 3 k +, m + α 2 m + α 3 m 2 + 2α 3 mk B 2mα 3 B 2.

9 The inverse matrix of some circulant matrices 9 Now adding all A + A 2 + A 3 = = n 2 6i n i + f k + α + α 2 k + α 3 k 2 f k α 2 + 2α 3 k K 2 kf k + α 3 K3 k 2 f k 2mα3 B 2 + α 2 m + α 3 m 2 + 2α 3 mk B = n 2 6i n i + α f k α 2 + 2α 3 kk 2 + α 3 K 3 2mα 3 B 2 + α 2 m + α 3 m 2 + 2α 3 mk B = d n a i V k q V m k q d 2c T m q c ai + c U k 2q + U m k q. Now observe that if k = then j =, and in this case A 2 = 0. Besides it holds s 2, m = f, s 2 2, m = K 2 f and s 3 2, m = K 3 f. Thus H i = g i, f + = g i,+l d f m+2 l n 2 6i n i + f α + α 2 m + + α 3 m + 2 f α 2 + 2α 3 m + K 2 f + α 3 K 3 f = n 2 6i n i + α α 2 m + α 3 m + 2 +α 2 + 2α 3 m + α 3 f α 2 + 2α 3 m + K 2 + α 3 K 3 = n 2 6i n i + α α 2 m α 3 m 2 f α 2 + 2α 3 m + K 2 + α 3 K 3 = 2di 6dn 2d 2 K 2 + 6d 2 K 3 = d n a i V m q du m q. 2c T m q Lemma 5. The matrix K is a block Toeplitz matrix given by K K 2... K m. K K = m K K2 K 2... K m K

10 0 S. Gago where each submatrix K k has as elements K k i,h+ = T m q 2cd a V m q d 2 U m q + kd n 2 h d 2 k 2, n h i + V k q V m k q + dh + nh nd n2 an c i h d nd U k 2 q + U m k q 6 + d2 6 with q = ad +, for any k =,..., m, i =,..., d and h = 0,..., d. 2c + 2 kd 3 d + 2 h n i Proof. Again we multiply two block Toeplitz matrices and thus K is a block Toeplitz matrix, and hence we can just compute only the first d rows. For each submatrix K k, k =,..., m, we compute its elements. We point out that the ih + -element of K k is the ij-element of K, with j = + k d + h, for any h = 0,..., d. K k i,h+ = h i,+l d g +l d,j = g i, h i, + h i,+l d g +l d,j + h i,+l d g +l d,j l= = A + A 2 + A 3. We compute separately A = h i, A 2 = = A 3 = = n 2 6h + kd dn h kd + d, h i,+l d n 2 6h + kd ldn h kd + ld h i,+l d n 2 6 h 2 2kdh k 2 d 2 + 2kd 2 + dhl d 2 l 2 + nh + kd ldn, h i,+l d n 2 6 h kd + ldn + h + kd ld, h i,+l d n 2 6 h 2 2kdh k 2 d 2 + 2kd 2 + dhl d 2 l 2 nh + kd + ldn. dk Besides, B = h i,+l d = a n 2c T m q U m q c a B 2 = V k q V m k q + V m q, dk k + 2 lh i,+l d = n a 2c T m q i U k 2 q + U m k q i k + U k 2 q + k + U m k q 2U m 2 q 2q V k q V m k q + V m q c kv k q kv m k q + V m q U k 2 q U m k q + U m q, a

11 The inverse matrix of some circulant matrices and K = h i,+l d = 0, l= K 2 = lh i,+l d = T m q V m q + i a d a 2c m + 2 m a 2c i m q + U m q m +, 2 K 3 = l 2 m + 3 h i,+l d = i + anm T m q d 2cd + an cd a V m q + 2c i m2 + 2m 3m q U m q + m2 2 + m + + 2c ad m + 2m +. 2 Now adding A + A 2 + A 3 = = n2 h i,+l d h i, 6h + kd dn h kd + d l= h i,+l d h 2 2kdh k 2 d 2 + 2kd 2 + dhl d 2 l 2 h i,+l d nh + kd ldn + h i,+l d nh + kd ldn h i, h + kd dn h kd + d + h 2 2kdh k 2 d 2 h i, = + 2kd 2 + dhk 2 h i, d 2 K 3 h i, + nh + kd2b + h i, dn2b 2 K 2 + h i, = 2nh i, kd + h d + 2nh + kdb 2ndB 2 + dn + 2dh + 2kd 2 K 2 d 2 K 3 n h i + V k q V m k q T m q an c i h d nd U k 2 q + U m k q 2cd a V m q d 2 U m q + dh + nh nd n2 6 + d kd 3 d + 2 h n i + kd n 2 h d 2 k 2, Finally, taking into account that qu m q U m 2 q = T m q, we obtain the desired result.

12 2 S. Gago Proposition 2. The group inverse of the Schur complement of L is a block Toeplitz matrix given by M M 2... M m. M M = m M M2 where each submatrix M k has as elements M k i,h+ = + T m q M 2... M m M i kd h n i kd h an n h i + V k q V m k q c i h d nd U k 2 q + U m k q 2cd a V m q d 2 U m q d 2 k 2 + d2, + dh + nh nd + 2 kd 3 d + 2 h n i + kd n 2 h with q = ad +, for any k =,..., m, i =,..., d and h = 0,..., d. 2c Lemma 6. The product Gs = n j m is a column vector where n i = a n6 + d + 5m + 6mii d 2, for any i =,..., d. Besides, it holds Ms = Gs. Proof. For a given row i =,..., d, we have Gs i = a g i,+k d = ag i, a g i,+k d = a n2 m k= k=2 i n i k d in k d + i a = a k=2 6n + 5m + dn 6mid + 2 i. Moreover, as G is a block Toeplitz matrix, it holds Gs i = Gs i+k d, for any k =,..., m. Besides, Ms = G GFGs = Gs GFGs = Gs, as it is straightforward to verify that FGs = 0,..., Acknowledgement This work has been supported by the Spanish Research Council Ministerio de Ciencia e Innovación under the project MTM R. References A. Ben-Israel, T.N.E. Greville:Generalized inverses. Theory and applications. 2nd Edition, Springer, New York, M.C. Golumbic, M. Stern, A. Levy, G. Morgenstern: Graph-Theoretic Concepts in Computer Science: 38th International Workshop, WG 202, Jerusalem, Israel, June 26-28, 202, Revised Selected Papers. Springer, 202 Edition.

13 The inverse matrix of some circulant matrices 3 3 E. Bendito, A. Carmona, A.M. Encinas, J.M. Gesto: A Formula for the Kirchhoff Index. Int. J. Quantum Chem., , E. Bendito, A. Carmona, A.M. Encinas, M. Mitjana: Generalized inverses of symmetric M matrices. Linear Algebra Appl., , A. Carmona, A.M. Encinas, M. Mitjana: Discrete elliptic operators and their Green operators. Linear Algebra Appl., , A. Carmona, A.M. Encinas, S. Gago, M.J. Jiménez, M. Mitjana: The inverse of some circulant matrices. Appl. Math. Comput., , A. Carmona, A.M. Encinas, S. Gago, M. Mitjana: Green operators of networks with a new vertex. Linear Algebra Appl., , Z. Cinkir: Deletion and contraction identities for the resistance values and the Kirchhoff index. Int. J. Quantum Chem., 20, C.M. da Fonseca, J. Petronilho: Explicit inverses of some tridiagonal matrices. Linear Algebra Appl., , P.J. Davis: Circulant Matrices. John Wiley & sons, New York, 979. A.M: Encinas, M.J. Jiménez: Floquet theory for second order linear homogeneous difference equations. J. Dif. Eq. and App., 206, DOI:0.080/ M.J. Jiménez: Ecuaciones en diferencias lineales de segundo orden y su aplicación al Análisis matricial. Phd. Thesis, G. Meurant: A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matrix Anal. Appl., 3 992, W. Wang, D. Yang, Y. Luo: The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs. Discrete Appl. Math., 6 203, M.A. Woodbury Inverting modified matrices, Statistical Research Group, Memo. Rep. no. 42, Princeton University, Princeton, N. J, 950.

Explicit inverse of a tridiagonal (p, r) Toeplitz matrix

Explicit inverse of a tridiagonal (p, r Toeplitz matrix A.M. Encinas, M.J. Jiménez Dpt. Matemàtiques, Universitat Politècnica de Catalunya - BarcelonaTech Abstract Tridiagonal matrices appears in many