Let $M$ and $N$ be two real square matrices of size $p+q$. The matrix $M$ is nonsingular. The matrix $N$ has the following block structure, where $A$ is a $q{\times}p$ matrix. $N = \left(\begin{array}{cc} \text{O}_{p,p} & \text{O}_{p,q} \\ A & \text{Id}_{q} \end{array}\right)$. I would like to conclude that the algebraic multiplicity of zero as an eigenvalue of the product matrix $MN$ is $p$. (The geometric multiplicity of the zero eigenvalue is preserved as $p$ from $N$ to $MN$.) I am looking for arguments to prove (or disprove) that conclusion.

up vote 3 down vote accepted

Take $p=q=1$, let $M = \begin{pmatrix} 0 & 1 \newline 1 & 0 \end{pmatrix}$ and let $N = \begin{pmatrix} 0 & 0 \newline 0 & 1 \end{pmatrix}$. Then the algebraic multiplicity of the zero eigenvalue in $MN = \begin{pmatrix} 0 & 1 \newline 0 & 0 \end{pmatrix}$ is $2$.

It's not true. Consider the case ($p = 2,\ q=1$) $N = \pmatrix{0 & 0 & 0\cr 0 & 0 & 0\cr a & b & 1\cr}$. The characteristic polynomial of $MN$ is ${t}^{3}- \left( am_{{13}}+bm_{{23}}+m_{{33}} \right) {t}^{2}$, and $a m_{13} + b m_{23} + m_{33}$ could easily be 0, resulting in an algebraic multiplicity of 3.

  • Thank you for this method of generating counterexamples. – Gilles Gnacadja Sep 20 '11 at 18:53

Your Answer

 
discard

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.