# A property of symmetric matrices

04/04/2011

**Fact**: Let be a symmetric matrix. Then for a vector , if and only if .

**Proof**:

Sufficiency: implies . Easy

Necessary: ???

**Wrong!!!** implies only holds for positive (semi) definite matrices! See Horn P400, problem 1 for proofs.

**EDIT**: if A is p.s.d, we can prove implies using SVD.

where . Then

implies . Therefore,

QED

If A is merely a symmetric matrix, we can’t have , right?

EDIT: for arbitrary matrix , we have

if and only if .

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