The vector space of commuting matrices

Task number: 2511

Let \(\mathbf D\) be a square matrix over a field \(\mathbb K\). Show that all the matrices which commute in matrix product with matrix \(\mathbf D\) form a vector space.

  • Hint

    Show that it is a subspace of the vector space of all square matrices over the field \(\mathbb K\).

  • Resolution

    All zero matrix commutes trivially with any matrix. Lets denote the set of all \(\mathbf D\)-commutable matrices as \(C\).

    Let \(\mathbf A, \mathbf B \in C\), then \((\mathbf A + \mathbf B)\cdot\mathbf D = \mathbf A\mathbf D + \mathbf B\mathbf D = \mathbf D\mathbf A + \mathbf D\mathbf B = \mathbf D\cdot(\mathbf A + \mathbf B)\)

    Now let \(\mathbf A \in C\) and \(a\in \mathbb K\), then \((a\mathbf A)\cdot\mathbf D = a(\mathbf A\cdot\mathbf D) = a(\mathbf D\cdot\mathbf A) = \mathbf D\cdot(a\mathbf A)\)

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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