Positive semidefinite but non diagonalizable real matrix - proof real parts of eigenvalues are non-negative

Question

I have a question about positive semidefinite matrices that are non diagonalizable. Example: \begin{equation} A= \left(\begin{array}{cc} 2 & 1\\ 0 & 2\\ \end{array}\right) \end{equation} Clearly the (real part of the) eigenvalues of $A$ are non-negative.

But how do I prove in general that the real part of the Eigenvalues of a positive semi-definite real matrix are non-negative? (I have seen the proof where they use diagonalization of the matrix ($B=T^{-1}DT$) but this is not possible for all positive semi-definite real matrices.)

Answer 1

I guess PSD matrices are symmetric and a symmetric matrix is orthogonally diagonalizable.We allways consider symmetry because otherwise eigen values can be complex and then it loses the essence of psd.