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 \((a\mathbf A)\cdot\mathbf D = a(\mathbf A\cdot\mathbf D) = a(\mathbf D\cdot\mathbf A) = \mathbf D\cdot(a\mathbf A)\)
