By Walter Borho, Peter Gabriel, Rudolf Rentschler
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2. (a) Leading principal submatrix. (b) Principal submatrix. 10. If A is an n X n positive definite Hermitian matrix, then every principal submatrix of A is also Hermitian and positive definite. In particular, the diagonal elements of A are positive. PROOF. Let Ap be any p x p principal submatrix. It is clear that Ap is Hermitian, because by deleting corresponding rows and columns of A to obtain Ap we maintain symmetry in the elements of Ap. Now let xp be any nonzero p-vector, and let x be the n-vector obtained from xp by inserting zeros in those positions corresponding to the rows deleted from A.
17) by formal matrix manipulations, but it is easier to argue as follows. Suppose that on the ith step of the elimination process there is a zero in the ith diagonal position, so that an interchange with, 22 Chapter I say, the kth row is necessary. Now imagine that we had interchanged the ith and kth rows of the original matrix A before the elimination began. Then when we arrive at the ith stage, we will find in the ith diagonal position exactly the nonzero element that we would have interchanged into that position in the original process.
If A is an n x n matrix, then A is nonsingular. Although this is true for some n x n matrices, it is false in general. 1. If A is nonsingular, it can be written in the form A = LV, where L is lower triangular and V is upper triangular. 2. If A == diag(A I , ••• , Ap), where each Ai is square, then A is nonsingular if and only if all Ai are nonsingular. 3. If x is a real or complex vector, then x T x is always nonnegative. 4. The Schur product of A and B is the same as the usual product when A and B have the same dimensions.
Primideale in Einhuellenden aufloesbarer Lie-Algebren by Walter Borho, Peter Gabriel, Rudolf Rentschler