The Centralizer of a Matrix is a Subspace

Vector Space Problems and Solutions

Problem 660

Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define
\[W = \{ A \in V \mid AM = MA \}.\] The set $W$ here is called the centralizer of $M$ in $V$.

Prove that $W$ is a subspace of $V$.

 
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Proof.

First we check that the zero element of $V$ lies in $W$. The zero element of $V$ is the $n \times n$ zero matrix $\mathbf{0}$.

It is clear that $M \mathbf{0} = \mathbf{0} = \mathbf{0} M$, and so $\mathbf{0} \in W$.


Next suppose $A, B \in W$ and $c \in \mathbb{R}$. Then $AM = MA$ and $BM = MB$, and so
\[( A + B ) M = A M + B M = M A + M B = M ( A + B ).\] Thus, $A + B \in W$.


We also have
\[( c A ) M = c ( A M ) = c ( M A ) = M ( c A ),\] and so $c A \in W$.

These three criteria show that $W$ is a subspace of $V$.


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