# Eigenvalues of the sum of a matrix such that all its principal submatrices are stable, and a diagonal matrix with non-positive entries

I have a real square matrix $$A$$ (not necessarily symmetric) with all its principal submatrices (including $$A$$) having eigenvalues with a negative real part.

On the other hand, I have a matrix $$D$$ which is diagonal and has only non-positive elements in its diagonal. I think that this implies that $$A+D$$ must also have eigenvalues with a negative real part. Does anyone know how to prove this, or if this is even true?

Principal minors of sum of a matrix and a diagonal matrix

That implies that the odd (resp. even) minors of $$A+D$$ will be negative (resp. positive). However, this doesn't prove my assertion.

Until now, I have only found the fact that $$A+D$$ cannot have a real positive eigenvalue, but could it have a complex-conjugate pair of eigenvalues with a positive real part?

EDIT: It is worth noticing that the condition on all principal submatrices to be stable is a requirement for the claim to be (possibly) true. In fact, if we only consider that $$A$$ is stable but we allow that some of its submatrices have eigenvalues with a positive real part, then the statement clearly does not hold. For instance, take $$A=\begin{pmatrix} 1 & -3 \\ 1 & -2 \end{pmatrix}.$$

This matrix is clearly stable. Nevertheless, if we consider the diagonal matrix

$$D=\begin{pmatrix} 0 & 0 \\ 0 & -3 \end{pmatrix},$$

then

$$A+D=\begin{pmatrix} 1 & -3 \\ 1 & -5 \end{pmatrix},$$

which is not a stable matrix. Notice that, in this case, $$A$$ has one principal submatrix with a positive real eigenvalue.

By going over Nonnegative Matrices in the Mathematical Sciences by Abraham Berman & Robert J. Plemmons, if we add that $$A \in \{ B = (b_{ij}) \in \mathbb{R}^{n \times n}: b_{ij}>=0 \text{ for } i \neq j \}$$, then this is an equivalence. It might hold for a larger class of matrices, but I could not find any proof that holds for the broader case.
Your description that "I have a real square matrix $$A$$ (not necessarily symmetric) with all its principal submatrices (including $$A$$) having eigenvalues with a negative real part." Leads to $$-A$$ having all principal minor being positive (A1 in the book).
If you go over [Berman & Plemmons, Ch.6 M-matrices] you will find that the description above means that $$-A$$ is a non-singular M-matrix. Which is also equivalent to existing a positive diagonal matrix $$P$$ such that $$-AP-PA^\top \succ 0$$ (H24 in the book.)
From there, given any $$D$$ which is diagonal and has only non-positive elements, since both $$-D$$ and $$P$$ matrices are non-negative diagonal matrices then $$-PD \succeq0$$, we have that $$(A+D)P+P(A+D)^\top = AP+PA^\top + 2DP \prec 2DP \preceq 0.$$ Therefore $$A+D$$ is Hurwitz stable.