A property of symmetric matrices
Fact: Let be a symmetric matrix. Then for a vector , if and only if .
Sufficiency: implies . Easy
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,
If A is merely a symmetric matrix, we can’t have , right?
EDIT: for arbitrary matrix , we have
if and only if .