Do the sum of a positive semi-definite matrix and a matrix which eigenvalues are positive or egal to zero have positive or egal to zero eigenvalues?

Question

Let's M be a symmetric positive semi-definite matrix and N a non symmetric matrix which all eigenvalues are positive or egal to zero.

The question is : Do the eigenvalues of the matrix M+N are all positive or egal to zero?

I have any clue for a proof, neither a counter example.

I write a small code that construct random matrices M and N with the good properties and check if the eigenvalues of M+N are positive or egal to zero. I let the code run for 5h and it found any counter example.

Any help would be appreciated.

Answer 1

Let $$ M = \left( \begin{array}{rr} 2 & -1 \\ -1 & 2 \\ \end{array} \right) $$ and $$ N = \left( \begin{array}{rr} 1 & -12 \\ 0 & 2 \\ \end{array} \right) $$ so that $$ M+N = \left( \begin{array}{rr} 3 & -13 \\ -1 & 4 \\ \end{array} \right) $$

Answer 2

Let's just elaborate on the example of @Will Jagy: take a positive definite matrix and add to it an upper triangular matrix with positive diagonal:

$$\begin{pmatrix} a&b\\b&d\end{pmatrix}+ \begin{pmatrix}1& t \\0 & 1 \end{pmatrix}$$

Its determinant is a linear function in $t$ with $-b$ the coefficient of $t$, so it can take any possible value, including negative ones.