1. Suppose that A E R"m is invertible. Show that if there exists a factorization A = LU where L is lower triangular with ones on the diagonal and U is upper triangular, then there is a unique such factorization. (Hint: what can you say about the inverse of a triangular matrix? What about the inverse of a triangular matrix with ones on the diagonal?) 2. Show that an arbitrary map ( • , • ) : Cri x Cm -+ C is an inner product on Cm if and only if there exists a Hermitian positive definite matrix B
E Cnxn such that (x, y) = y*Bx for all x, y E Cn.
3. Suppose that A, B E Rmxm are positive definite matrices. Show that AB is diagonalizable with strictly positive eigenvalues. 4. Suppose that X, Y E Rmxk where k < n and XX* = YY*. Show that there exists an orthogonal matrix Q E Rkxk such that Y= XQ. 5. Let A E Cm" be a Hermitian matrix, and fix the Euclidean inner product / norm on C. Let Al < • • • < An be the eigenvalues of A, repeated according to multiplicity. Show that
Ai = inf fu*Au : liull = 11.
More generally show that if ui, ... , uj are orthonormal eigenvectors with eigenvalues A1, ... , Ak, re-spectively, where k < n, then
Ak±i = inf {u*Au : Ilull = 1, (u,ui) = 0 for all j = 1...,k} .
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