Assuming I have a complex non-symmetric matrix $A$ which is "positive definite" in the sense that $\Re(x^*Ax) > 0$.
A necessary and sufficient condition for $A$ to be "positive definite" is that the Hermitian part $A_H = A+A^*$ is positive definite (A^* is the conjugate transpose). (Source Wolfram Alpha)
Now I am wondering: is it also necessary and/or sufficient that the Hermitian matrix $A_{H2} = AA^*$ is positive definite?
Somehow I think that I have seen this somewhere before, and it would be really useful if it were true. ;)
Thanks a lot for your help!
Cheers, Christoph
$\endgroup$1 Answer
$\begingroup$$AA^*$ is always positive semidefinite, and positive definite iff $A$ is invertible. Which is far weaker than any form of positive definiteness of $A$ itself. (Take $A=-I$ for an extreme example.)
$\endgroup$ 3
