Tikrit Journal of Pure Science

I n this study the inverse of two patterned matrices has been investigated. First, for a Toeplitz-type matrix, it is proved that the exact number of independent cofactors is (n +2)/4 when n is even number and (𝒏 + 𝟏) 𝟐 /𝟒 when n is an odd. Second, when the matrix is reduced to a Jacobi-type matrix B n , two equivalent formulae for its determinant are obtained, one of which in terms of the eigen values. Moreover, it is proved that the independent cofactors 𝑩 𝒊𝒋 of 𝑩 𝒏 are explicitly expressed as a product of the determinants of 𝑩 𝒊−𝟏 and 𝑩 𝒏−𝒋 . So, the problem of finding the exact inverse of 𝑩 𝒏 is reduced to that one of finding the determinants of 𝑩 𝒊 , i = 1, 2, …, n.


1-Introduction
One of the important problems involved in the analysis of such models is to find the exact inverse of these covariance matrices in explicit form which leads to the computation of determinants and other related characteristics such as their eigen values and spectral representation.Such computations are tedious especially when the order n of the matrix is large [2].There is a large literature on inversion of covariance matrices (e.g.[3,2,4]).The problem has been approached either numerically to find fast algorithms or analytically to find explicit forms for the entries of the inverse.Naturally, analytical solution leads to numerical one.Now, let   be an (n x n) symmetric, positive definite matrix.  is said to be a patterned matrix if its entries exhibit a structured form, for example the Toeplitz matrix, the Jacobi matrix, ….These patterned matrices are frequently encountered as covariance matrices of structured dependent errors or observations in statistical models or autoregressive and moving average time series models as well as in many other stochastic models [1].The purpose of this work is divided to two parts.We first prove for a Toeplitz-type matrix that the number of independent cofactors is exactly ( + 2) 4 ⁄ for n even and ( + 1) 2

4
⁄ for n odd.This reduces the number of distinct cofactors to a little bit greater than the quarter of the total number  2 of cofactors, which means that, practically, only these distinct entries of the adjoint matrix need to be calculated [5].Further, these distinct elements have a certain arrangement along each diagonal on the upper half of the matrix.Second when the matrix is reduced to a Jacobi-type matrix   , two equivalent formulae for the determinant of   are given, one of them in terms of the eigen values of the matrix.Moreover, it is proved that the independent cofactors   of   are exactly given by: When n is odd or n/2 when n is even, and b is some entry of   .So that the problem of finding the inverse of a Jacobimatrix is reduced to that of finding the determinant of   , i = 1,2, …, n. [6] 2-The Adjoint of A Toeplitz-type matrix: Suppose   = [  ] is a Toeplitz matrix of order n having the form:   =  − , 1 ≤ ,  ≤ .[4] Let   denote the submatrix of order n-1 obtained by deleting the ith row and the jth column of   , and let   = (−1) + (  ) be the cofactor of   .It is well-known that the inverse   −1 of   is given by: , where t denote the transpose of the matrix = [  ], by symmetry of   .This means that   for all i > j are redundant.The following lemma proves that about the half of the remaining cofactors are redundant too.
Now, let n be even, n = 2r with r a positive integer.Then the independent cofactors are those   with i = 1,2, …, r, i j  n-i+1, so that The last summation can be proved to be exactly (  −   ) 2 −2 , which leads to the expression: ] the greatest integer ≤  2 … (3.3)

3-2 The Eigen Values of 𝑩 𝒏 :
If  is an Eigen value of   if  satisfies the linear equation    = , with Z a nonzero column vector of dimension n, which is the Eigen vector corresponding to .To find the Eigen values of   we are motivated by the approach relating to this problem to the characteristic-value problem of a finite homogeneous boundary difference system of equations (See [9]).In theorem 3.1 below we prove that the n eigen values of   are exactly the ne eigen values of a system of ne difference equtions with two boundary conditions, and hence can be determined from the general solution of the system.= z m …………………………………… bz n-1 + az n = z n which is equivalent to the homogeneous system of different equations bz m+1 + az m + bz m-1 = z m , m = 1,2, … , n. with the two homogeneous boundary conditions z 0 = 0 , z m+1 = 0 .For such a system, no nonzero solution exists unless  takes on one of a set of eigen values  1 , …,  n which are exactly the required eigen values of B n .In fact no nonzero solution to the above system exists unless | − 2 | < 1 or equivalently unless  = a -2b cos .In this case the general solution to the system is z m = c 1 cos m .The condition z 0 = 0 implies c 1 = 0 , and the second condition z m+1 = 0 leads to c 2 sin (n+1) = 0, which unless  takes a value for which sin(n+1) = 0, the only solution is c 2 = 0, in which case z m = 0, m = 1,2, … , n.However, if (n+1 which is another expression of det (B n ).

3-3 The Inverse of 𝑩 𝒏 :
As proved in theorem 2.1, to find adj B n it suffices to calculate the cofactors B ij , j = i, i+1, …., n-i+1, i = 1, 2, …, for n even.Observe that when deleting the ith row of B n , for any fixed i , the obtained submatrix gives the following cofactors B ij , j = i, i+1, …, n-i+1, where, B ij = (-1) 2i det (B i-1 ) det (B n-i ) , j = i , … (3.5) B ij = (-1) i+j det (C ij ) det (B n-j ), j = i+1, … , n-i+1 with C ij a square matrix of order (j-1) satisfying the relation: as n odd or even.Clearly, this formula reflects the symmetry of cofactors proved before for the more general case by lemma 2.1.We can thus state the theorem: Theorem 3.2: For the matrix   given in (3.1), the independent cofactors   are exactly: as n odd or even.Remark 3.1: 1-It follows from theorem 3.2 that, to find  −1 it suffices to calculate the determinants of B 1 , B 2 , … B n which can be calculated using either formula (3.3) or (3.4).2-A statement similar to that of B ij in the theorem but for the inverse of the covariance matrix of a first order moving average process has been observed before by Arato [7] and then used shaman [10].

4-Applications
Below are two examples of statistical models for which the involved covariance matrix is of the Toeplitz of Jacobi types studied in this work.

4 Remark 2 . 1 1 2 ) 2
The independent cofactors are exactly the entries of the adjoint matrix indicated by the hachured area n odd or even, respectively.[7]3-The Jacobi-type matrix:[8] In all this section we suppose that the matrix = [ |−| ] is now reduced to a Jacobi-type matrix where  |−| =0 whenever | − | > 1. Precisely, we suppose a matrix   = [  ] such that : The determinant of   : Let   = (  ).Then by expansion about the first column, it can be shown that   satisfies the difference equation of second order:   =  −1 −  2  −2 , n = 2,3, … with the two boundary conditions  0 = 1,  1 = 0.The roots of the auxiliary equation  2 −  +  2 = 0 are, Expanding the binominals in (3.2),   reduced to : Expansion of the above binomial again yields.