(adsbygoogle = window.adsbygoogle || []).push({}); 1. Let A[itex]\in[/itex]R^{nxn}be a symmetric matrix, and assume that there exists a matrix B[itex]\in[/itex]R^{mxn}such that A=B^{T}B.

a) Show that A is positive semidefinite

B) Show that if B has full rank, then A is positive definite

2. Relevant equations:

This is for an operations research class, so most of the definitions revolve around minimizing/maximizing.

However, alternate definitions:

Positive Definite: A is positive definite if for all non-null vectors h, h^{T}Ah > 0.

Symmetric: if A^{T}=A.

Semidefinite: h^{T}Ah ≥ 0

3. The attempt at a solution

Here's some work:

A^{T}= A ; A = B^{T}B.

So, A^{T}= B^{T}B → A^{T}A = B^{T}BA = AA = A^{2}.

So, A^{T}A ≥ 0.

But, that's not quite what I want.

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# Homework Help: Linear Algebra: Symmetric/Positive Definite problem

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