For this section the following two definitions will be relevant for an n x n matrix A. • A is positive iff for all æ, x# Ax is real and non-negative. • A is positive-definite iff A is positive and x# Ah = 0 iff x = 0. Problem 1. Use the Spectral Theorem to show that • A is positive and Hermitian iff A = B#B_for some matrix B. • A is positive definite and Hermitian iff A = B#B for some B with NS(B) = {0}. In some sense B is the correct notion of the square-root of A.

For this section the following two definitions will be relevant for an n x n matrix A. • A is positive iff for all æ, x# Ax is real and non-negative. • A is positive-definite iff A is positive and x# Ah = 0 iff x = 0. Problem 1. Use the Spectral Theorem to show that • A is positive and Hermitian iff A = B#B_for some matrix B. • A is positive definite and Hermitian iff A = B#B for some B with NS(B) = {0}. In some sense B is the correct notion of the square-root of A.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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