I have read that the singular values of any matrix $A$ are non-negative (e.g. wikipedia).
Is there a reason why?
The first possible step to get the SVD of a matrix $A$ is to compute $A^{T}A$. Then the singular values are the square root of the eigenvalues of $A^{T}A$. The matrix $A^{T}A$ is a symmetric matrix for sure. The eigenvalues of symmetric matrices are always real. But why are the eigenvalues (or the singular values) in this case always non-negative as well?
Best Answer
I'm assuming that the matrix $A$ has real entries, or else you should be considering $A^*A$ instead.
If $A$ has real entries then $A^TA$ is positive semidefinite, since $$ \langle A^TAv,v\rangle=\langle Av,Av\rangle\geq 0$$ for all $v$. Therefore the eigenvalues of $A^TA$ are non-negative.