By Jørn Børling Olsson

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Structure and Representation of Jordan Algebras

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As an application of the Fischer inequality, we give a determinantal inequality. Let A, 5 , C, D be square matrices of the same size, so that A C B \ f A* D J \ B* C ' * \ / AA* + 5 5 * AC* + BD'' \ D"" J ~ \ CA"" H- L>5* CC* -f DD* ) Then det( ^ ^"jl < d e t ( A A * + 5 5 * ) d e t ( C C * + L>J9*). - SEC. 4 POSITIVE SEMIDEFINITE MATRICES 35 If A and C commute, then I det(^i:> - CB)\^ < det(AA* + ^ 5 * ) det(CC* + DD""). The Fischer inequahty and an induction gives the celebrated Hadamard inequality.

M} and {1, 2 , . . , n}, respectively. ]A[a,^]^A[a,/3^]. 36) It is usually convenient to think of A [a, f3] as being in the upper left corner of A (not necessarily square), a placement that can always be achieved with suitable row and column permutations, that is, with permutation matrices P and Q such that li a — p and m = n, A [a, /3] is a principal submatrix of A and P = Q^. 36) with an unspecified generalized inverse, we would have to impose conditions sufficient to ensure that the generalized Schur complement obtained in this way did not depend on the choice of the generalized inverse.

Let A and B be square matrices of orders n and ra, respectively, with n > m. If there is a solution X of rank m of the homogeneous linear matrix equation AX — XB = 0, it is known that the m eigenvalues of B are also eigenvalues of A. The following theorem exhibits a matrix (a Schur complement) whose eigenvalues are the remaining n — m eigenvalues of A. 10 Suppose that n > m and let A e C^""^ and B G C ^ ^ ^ . Let X G C^^"^ be such that AX — XB, partition X and A conformally as and assume that Xi is m x m and nonsingular.