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How can I get to know a basis of a vector-subspace of $\mathbb{R}^{2 \times 2}$ formed by matrices $X$ that commute with the matrix: $$A=\left[\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right]$$

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Take an arbitrary matrix $X \in \mathbb{R}^{2\times 2}$ and compute the left and right products with $A$. This should give you restrictions on the coefficients in the matrix.

Say $$X=\left[\begin{matrix}x_{11} & x_{12} \\ x_{21} & x_{22} \end{matrix}\right] \; .$$

Then $XA=AX$ implies that $x_{21}=0$ and $x_{11}=x_{22}$. From this, it is straightforward to build a basis for the space of matrices $X$.

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Inputs $$ \mathbf{A} = \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right), \qquad \mathbf{X} = \left( \begin{array}{cc} 0 & 1 \\ 0 & 0 \\ \end{array} \right) $$

Multiplication $$ \mathbf{AX} = \left( \begin{array}{cc} 0 & a \\ 0 & c \\ \end{array} \right), \qquad \mathbf{XA} = \left( \begin{array}{cc} c & d \\ 0 & 0 \\ \end{array} \right) \tag{1} $$

Conditions

Under what conditions does $$ \begin{align} \mathbf{AX} &= \mathbf{XA} ?\\ \left( \begin{array}{cc} 0 & a \\ 0 & c \\ \end{array} \right) &= \left( \begin{array}{cc} c & d \\ 0 & 0 \\ \end{array} \right) \end{align} $$

Conditions to satisfy $(1)$:

  1. $a=d$

  2. $c=0$

  3. $b\in\mathbb{C}$

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