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I wonder if someone can prove/disprove the following inequality,

$\lambda_i(A+mI) \leq \lambda_i(A+K) \leq \lambda_i(A+MI)$

where $A$ is a real symmetric Metzler matrix with real and nonpositive eigenvalues and $K$ is a diagonal matrix. $M$ and $m$ are the greatest and the least elements of $K$ respectively.

I have seen that the above inequality holds for all examples I tried but I would like to prove it mathematically or find a counterexample.

Any help would be appreciated. Thanks.

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  • $\begingroup$ Are you aware of the fact that $\lambda_i(A+xI)=\lambda_i(A)+x$? $\endgroup$
    – Taneli Huuskonen
    Commented Mar 9, 2018 at 22:47
  • $\begingroup$ Yes, for sure ! $\endgroup$
    – Mohammad
    Commented Mar 9, 2018 at 23:25
  • $\begingroup$ But how does that help? $\endgroup$
    – Mohammad
    Commented Mar 10, 2018 at 2:19
  • $\begingroup$ It shows that you can add any real multiple of $I$ to $A$ or $K$ (and change $m$ and $M$ accordingly) in the question, and what you get is equivalent to the original claim. So, the assumption about the eigenvalues of $A$ is irrelevant. It also suffices to prove the inequality $\lambda_i(A)\le \lambda_i(A+K)$ whenever $K$ is a real diagonal matrix with nonnegative entries; the second one then follows since $A+MI=(A+K)+(MI-K)$. $\endgroup$
    – Taneli Huuskonen
    Commented Mar 10, 2018 at 18:50

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This follows from the Interlacing Eigenvalue Theorem (Golub and Van Loan "Matrix Computations", 4th edition, Theorem 8.1.8), and holds for any real symmetric n by n $A$, whether or not Metzler or having nonpositive eigenvalues.

$A + K = A + mI +$ sum of n nonnegative multiples of rank one matrices of the form $cc^T$ for $||c||_2 = 1$. The ith rank one matrix can be taken as $e_ie_i^T$, where $e_i$ is the ith unit vector, with multiple $(K_{ii}-m)$. Interlacing holds as each successive nonnegative multiple of rank one matrix is added, and therefore for $A + mI$ vs. $A + K$. Similarly for $A + K$ vs. $A + MI$.

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