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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.

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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 $N = \left(\begin{array}{cc} \textnormal{O}{p,p} text{O}_{p,p} & \textnormal{O}{p,q} text{O}_{p,q} \\ A & \textnormal{Id}_{q} text{Id}_{q} \end{array}\right)$' 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.

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# Preserving the algebraic multiplicity of the zero eigenvalue

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. '$N = \left(\begin{array}{cc} \textnormal{O}{p,p} & \textnormal{O}{p,q} \ A & \textnormal{Id}_{q} \end{array}\right)$' 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.