[Math] How to you expand the adjoint of a matrix into a polynomial with matrix coefficients

Question

This book contains an algorithm which claims that a matrix $sI – A$, where $A$ is some $n \times n $ square matrix and $s$ a variable can be expanded into

$$adj(sI – A) = K_0 s^{n-1} + K_1 s^{n-2} +…+K_{n-1}$$

where the $K_i$s are matrices.

Of course when I first read it I thought it was completely crazy. For example, given $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$, the adjoint is $adj(A) = \begin{bmatrix} 4 & -3 \\ -2 & 1 \end{bmatrix}$ obviously not a polynomial…

But then I realized the $K_is$ are matrices, so if one can pull out the $s$ somehow, then the adjoint matrix $adj(sI-A)$ might just turn into a polynomial with matrix coefficients.

Does anyone know how to achieve this? Is there a well known method?

Answer 1

Well, there is explanation in the book at the same page:

Note that since each element of $\Gamma(s)$ (the adjoint matrix of $sI-A$) is obtained from the determinant of $(n-1)\times(n-1)$ matrix, each element $\Gamma_{ij}(s)$ is a polynomial in $s$ of degree no greater than $s^{n-1}$

And since we can group terms with the same degree and put the common multiplier before the grouped terms, the decomposition of ${\rm adj}\: (sI-A)$ as $K_{n-1} s^{n-1} + K_{n-2} s^{n-2} + \dots + K_1 s + K_0, \; K_i \in \mathbb{R}^{n \times n}$ is quite straightforward.