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A property of symmetric matrices


Fact: Let A be a symmetric matrix. Then for a vector x, x^TAx=0 if and only if Ax=0.


Sufficiency: Ax=0 implies x^TAx=0. Easy

Necessary: ???

Wrong!!! x^TAx=0 implies Ax=0 only holds for positive (semi) definite matrices! See Horn P400, problem 1 for proofs.

EDIT: if A is p.s.d, we can prove x^TAx=0 implies Ax=0 using SVD.

A=U\Sigma U^T=U\sqrt{\Sigma}\sqrt{\Sigma}^TU^T=B^TB

where B=\sqrt{\Sigma}^TU. Then

x^T A x=x^TB^TBx=y^Ty=0

implies y=Bx=0. Therefore,

Ax=B^TBx=0 QED

If A is merely a symmetric matrix, we can’t have A=U\Sigma U^T, right?


EDIT: for arbitrary matrix A, we have

A^TAx=0 if and only if Ax=0.

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