Let $A$ be a $m\times n$ matrix with complex entries and let $A^*$ be it's conjugate transpose , then off-course $A^*A$ is a Hermitian matrix whence all its eigenvalues are real ; is it also true that all the eigenvalues of $A^*A$ are non-negative ?
[Math] All the eigenvalues of $A^*A$ are non-negative
eigenvalues-eigenvectorsmatrices
Best Answer
Let's see: if $\lambda$ is an eigenvalue of $A^*A$ then for some nonzero $x$, $$ \lambda \|x\|^2 = \langle \lambda x,x \rangle = \langle A^*A x,x\rangle = \langle Ax,Ax \rangle \ge 0.$$