# If M = ((0,0,-1),(1,0,-1),(0,1,0)) and A is an invertible rational 3xx3 matrix which commutes with M, then is A necessarily expressible as A = aM^2+bM+cI_3 for some scalar factors a, b, c?

## This $M$ is the companion matrix of the polynomial ${x}^{3} + x + 1$. It satisfies: ${M}^{3} + M + {I}_{3} = 0$ and this is its minimum polynomial. Hence the identity matrix ${I}_{3}$ with $M$ generates a field of matrices all expressible in the form $a {M}^{2} + b M + c {I}_{3}$.

Feb 22, 2017

Yes...

#### Explanation:

Given:

$M = \left(\begin{matrix}0 & 0 & - 1 \\ 1 & 0 & - 1 \\ 0 & 1 & 0\end{matrix}\right)$

Then:

${M}^{2} = \left(\begin{matrix}0 & 0 & - 1 \\ 1 & 0 & - 1 \\ 0 & 1 & 0\end{matrix}\right) \left(\begin{matrix}0 & 0 & - 1 \\ 1 & 0 & - 1 \\ 0 & 1 & 0\end{matrix}\right) = \left(\begin{matrix}0 & - 1 & 0 \\ 0 & - 1 & - 1 \\ 1 & 0 & - 1\end{matrix}\right)$

Suppose $A = \left(\begin{matrix}c & u & v \\ b & w & x \\ a & y & z\end{matrix}\right)$ is a $3 \times 3$ matrix that commutes with $M$.

We have:

$A M = \left(\begin{matrix}c & u & v \\ b & w & x \\ a & y & z\end{matrix}\right) \left(\begin{matrix}0 & 0 & - 1 \\ 1 & 0 & - 1 \\ 0 & 1 & 0\end{matrix}\right) = \left(\begin{matrix}u & v & - c - u \\ w & x & - b - w \\ y & z & - a - y\end{matrix}\right)$

$M A = \left(\begin{matrix}0 & 0 & - 1 \\ 1 & 0 & - 1 \\ 0 & 1 & 0\end{matrix}\right) \left(\begin{matrix}c & u & v \\ b & w & x \\ a & y & z\end{matrix}\right) = \left(\begin{matrix}- a & - y & - z \\ c - a & u - y & v - z \\ b & w & x\end{matrix}\right)$

Equating the elements of $A M$ and $M A$, we find:

$u = - a$

$w = c - a$

$y = b$

$v = - y = - b$

$x = - a - y = - a - b$

$z = w = c - a$

So:

$A = \left(\begin{matrix}c & - a & - b \\ b & c - a & - a - b \\ a & b & c - a\end{matrix}\right)$

$\textcolor{w h i t e}{A} = a \left(\begin{matrix}0 & - 1 & 0 \\ 0 & - 1 & - 1 \\ 1 & 0 & - 1\end{matrix}\right) + b \left(\begin{matrix}0 & 0 & - 1 \\ 1 & 0 & - 1 \\ 0 & 1 & 0\end{matrix}\right) + c \left(\begin{matrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{matrix}\right)$

$\textcolor{w h i t e}{A} = a {M}^{2} + b M + c {I}_{3}$