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Combined Matrices of Matrices in Special Classes

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Combined Matrices of Matrices in Special Classes Combined matrices of matrices in special classes Miroslav Fiedler Institute of Computer Science AS CR May 2010 Combined matrices of matrices in special classes Let A be a nonsingular (complex, or even more general) matrix. In [3], we called the Hadamard (entrywise) product A ± (A¡1)T the combined matrix of A, and denoted it as C(A). It has good properties: All its row- and column-sums are equal to one. If we multiply A by a nonsingular diagonal matrix from the left or from the right, C(A) will not change. It is a long standing problem to characterize the range of C(A) if A runs through the set of all positive de¯nite n £ n matrices. A partial answer describing the diagonal entries of C(A) in this case was given in [2]: Theorem 1. A necessary and su±cient condition that the numbers a11, a22, : : : ; ann are diagonal entries of a positive de¯nite matrix and ®11; ®22;:::, ®nn the diagonal entries of its inverse matrix, is ful¯lling of : 1. aii > 0, ®ii > 0 for all i, 2. a ® ¡ 1 ¸ 0 for all i, ii ii p P p 3. 2 maxi( aii®ii ¡ 1) · i( aii®ii ¡ 1): In the notation above, it means that a matrix M = [mik] can serve as C(A) for A positive de¯nite, only if its diagonal entries satisfy m ¸ 1 and p P p ii 2 maxi( mii ¡ 1) · i( mii ¡ 1): In fact, there is also another necessary condition, namely, that the matrix M ¡ I, I being the identity matrix, is positive semide¯nite. Let us show that, for simplicity in the real case. We have for the ¡1 quadratic form A ± A if x = (x1; : : : ; xn); tr meaning the trace and X the diagonal matrix X = diag(xi); Xn Xn aik®kixixk = aikxk®kixi i;k=1 i;k=1 = trAXA¡1X: Since A is positive de¯nite, it can be written as LLT ; where L is a nonsingular lower triangular matrix. Thus the last expression is trLLT X(LT )¡1L¡1X; which again is equal to trLT X(LT )¡1L¡1XL; or trZT Z; where Z = L¡1XL: The matrix Z is lower triangular and its diagonal is X. Since trZT Z is the sum of squares of all the entries of Z, it is greater than or equal to P 2 Pn P 2 i xi : Altogether, i;k=1 aik®kixixk ¸ i xi : Let us compare the conditions in Theorem 1 with the fact that M ¡ I is positive semide¯nite. We already saw that M has all row- (and column-) sums equal to one, i. e. Me = e; where e is the vector of all ones. Since M ¡ I is positive semide¯nite, it is Gram matrix of some vectors u1; : : : ; un in a Euclidean space, and by Me = e, the sum of these vectors is the zero vector. It follows that they can be considered as vectors forming edges of a closed polygon so that each of the vectors has length not exceeding the sum of lengths of the remaining vectors: X juij · jukj; k;k6=i P P i.e. 2ju j · ju j; i.e. 2 max ju j · ju j: i pk k i i i i Since juij = aii®ii ¡ 1; we obtained that p X p 2 max aii®ii ¡ 1 · aii®ii ¡ 1: i i However, this inequality is weaker than that in 3: of Theorem 1. In [1], a similar problem was considered for the case that A is an n £ n M-matrix, i.e. a real matrix all o®-diagonal entries of whose are non-positive but such that for some positive vector u, Au is also positive. The inverse of such matrix exists and is a nonnegative matrix. This implies that C(A) has also all o®-diagonal entries non-positive and since Ae > 0, the combined matrix of an M-matrix is also an M-matrix. The matrix C(A) ¡ I is then a "singular M-matrix"; it can be obtained as a singular limit of a convergent sequence of M-matrices. We showed that the diagonal entries mii of C(A) are characterized by the conditions 1. mii ¸ 1 for all i, and P 2. 2 maxi(mii ¡ 1) · i(mii ¡ 1): Apparently, it is not known whether every M-matrix with row- and column-sums one can serve as combined matrix of some M-matrix A. Theorem 2. Let A = [aik] be an arbitrary (even complex) ¡1 nonsingular tridiagonal matrix, A = [®ik]. Then the sequence faii®ii ¡ 1g has the property that the sum of its odd terms is equal to the sum of the even terms. Proof. Let us form the combined matrix C = A ± (AT )¡1. Since all its row and column sums are equal to one, the matrix C ¡ I has all such sums equal to zero. It is also tridiagonal. Let now ro, re, respectively, be the sum of all entries in the odd, respectively even rows of C ¡ I, and analogously co, ce the sums of the columns of C ¡ I. It is evident that the sum ro ¡ re + co ¡ ce, which is zero, is at the same time twice the alternate sum of the diagonal entries of C ¡ I. A symmetric real Cauchy matrix A is an n £ n matrix assigned to n real parameters x1; : : : ; xn, as follows: · ¸ 1 A = ; i; j = 1; : : : ; n: xi + xj It is well known that A is nonsingular if and only if, in addition to the general existence assumption that xi + xj 6= 0 for all i and j, the xi's are mutually distinct. In fact, there is a formula for the determinant of A Q 2 i;k;i>k(xi ¡ xk) det A = Qn : (1) i;j=1(xi + xj) It is easily seen that such Cauchy matrix is positive de¯nite if all the xi's are positive. If they are ordered, say x1 > x2 > ¢ ¢ ¢ > xn > 0; then A is totally positive, i.e. all square submatrices of A have positive determinant. We will again be interested in the combined matrix of C, in particular, in the sequence of its diagonal entries fmiig; by proving: ½ p p p p 1 for n odd; m ¡ m + m ¡+ ¢ ¢ ¢+(¡1)n¡1 m = 11 22 33 nn 0 for n even: (2) We prove ¯rst: Observation Let A = [aij] be a nonsingular symmetric Cauchy matrix, aij = 1=(xi + xj): Then there exists a diagonal nonsingular matrix D, such that A¡1 = DAT D: (3) To prove that, let X = diag(x1; : : : ; xn): Then, since xiaij + aijxj = 1 for all i; j, XA + AX = J; the matrix of all ones. Multiply this by A¡1 from both sides. We obtain XA¡1 + A¡1X = A¡1eeT A¡1: ¡1 T If we denote the vector A e as (z1; : : : ; zn) and the entries of ¡1 A as ®ij, the previous equation can be written as xi®ij + ®ijxj = zizj; i.e. (3): zizj ®ij = xi + xj as asserted. Also, the zi's are all di®erent from zero. Suppose now that A is totally positive. It is well known that the inverse has then the checkerboard sign pattern, which means that the matrix D will have all positive diagonal entries (or, all negative) if multiplied by the diagonal matrix S = diag(1; ¡1; 1;:::; (¡1)n¡1): Let L = DS be this diagonal matrix with positive diagonal. Thus A¡1 = LSASL; i.e. ALSASL = I; which can be written as 1 1 1 1 L 2 AL 2 SL 2 AL 2 S = I: 1 1 The matrix Q = L 2 AL 2 has thus the property that (QS)¡1 = QS; i.e., QS is involutory. Therefore, the eigenvalues of QS are 1 and ¡1 only. On the other 1 1 hand, Q is positive de¯nite, so that the matrix Q 2 SQ 2 which has the same eigenvalues as QS is congruent to S. Thus the eigenvalues of QS are 1 and ¡1 with the same multiplicity if n is even, the multiplicity of 1 being greater by one if n is odd. Now, if ¡1 q11; q22; : : : ; qnn are the common diagonal entries of Q and Q , n¡1 then q11; ¡q22;:::; (¡1) qnn are the diagonal entries of QS which implies that trace of QS is 0 or 1. Since the combined matrices C(A) and C(Q) are the same, we have, if mii are their 2 diagonal entries (i.e., mii = qii) ½ p p p p 1 for n odd; m ¡ m + m ¡+ ¢ ¢ ¢+(¡1)n¡1 m = 11 22 33 nn 0 for n even; as we wanted to prove. This result can be applied to every principal submatrix of the Hilbert matrix 2 1 1 1 1 3 1 2 3 4 ¢ ¢ ¢ n 6 1 1 1 1 ¢ ¢ ¢ 1 7 6 2 3 4 5 n+1 7 6 1 1 1 1 1 7 6 3 4 5 6 ¢ ¢ ¢ n+2 7 ; 4 ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢ ¢¢¢ 5 1 1 1 1 1 n n+1 n+2 n+3 ¢ ¢ ¢ 2n¡1 or, · ¸ 1 H = ; i; k = 1; : : : ; n: n i + k ¡ 1 1 since it is a totally positive Cauchy matrix (choose xi = i ¡ 2 ). Let us show it on an example. Example. The submatrix 2 1 1 1 3 3 4 5 4 1 1 1 5 A = 4 5 6 1 1 1 5 6 7 of the Hilbert matrix has the inverse 2 3 300 ¡900 630 4 ¡900 2880 ¡2100 5 : 630 ¡2100 1575 Thus 2 3 100 ¡225 126 A ± A¡1 = 4 ¡225 576 ¡350 5 126 ¡350 225 is an integral matrix.
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