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**1. Let A[itex]\in[/itex]R**

a) Show that A is positive semidefinite

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

^{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. Homework 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

## 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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