Characteristic Polynomial and Minimum Polynomial

The characteristic polynomial of a square matrix A is the determinant of the characteristic matrix xI - A.

$\blacktriangleright$ Matrices + Characteristic Polynomial

$\left(\vphantom{
\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}
}\right.$$\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}
}\right)$, characteristic polynomial: $\left(\vphantom{ X-4}\right.$X - 4$\left.\vphantom{ X-4}\right)^{{3}}_{}$

$\blacktriangleright$ Evaluate

X$\left(\vphantom{
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
}\right.$$\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}
}\right)$ - $\left(\vphantom{
\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}
}\right.$$\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}
}\right)$ =  $\left(\vphantom{
\begin{array}{ccc}
-4+X & -1 & 0 \\
0 & -4+X & 0 \\
0 & 0 & -4+X
\end{array}
}\right.$$\begin{array}{ccc}
-4+X & -1 & 0 \\
0 & -4+X & 0 \\
0 & 0 & -4+X
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
-4+X & -1 & 0 \\
0 & -4+X & 0 \\
0 & 0 & -4+X
\end{array}
}\right)$

det $\left(\vphantom{
\begin{array}{ccc}
-4+X & -1 & 0 \\
0 & -4+X & 0 \\
0 & 0 & -4+X
\end{array}
}\right.$$\begin{array}{ccc}
-4+X & -1 & 0 \\
0 & -4+X & 0 \\
0 & 0 & -4+X
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
-4+X & -1 & 0 \\
0 & -4+X & 0 \\
0 & 0 & -4+X
\end{array}
}\right)$ =  $\left(\vphantom{ X-4}\right.$X - 4$\left.\vphantom{ X-4}\right)^{{3}}_{}$

The minimum polynomial of a square matrix A is the monic polynomial p(x) of smallest degree such that p(A) = 0. By the Cayley–Hamilton theorem , f (A) = 0 if f (x) is the characteristic polynomial of A. The minimum polynomial of A is a factor of the characteristic polynomial of A.

$\blacktriangleright$ Matrices + Minimum Polynomial, Factor

$\left(\vphantom{
\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}
}\right.$$\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}$$\left.\vphantom{
\begin{array}{ccc}
4 & 1 & 0 \\
0 & 4 & 0 \\
0 & 0 & 4
\end{array}
}\right)$, minimum polynomial: 16 - 8X + X2 =  $\left(\vphantom{ X-4}\right.$X - 4$\left.\vphantom{ X-4}\right)^{{2}}_{}$



\begin{example}
This example illustrates the Cayley--Hamilton theorem.
\par
Def...
... & 0 & 0 \\
0 & 0 & 0
\end{array}
\right)
\end{displaymath}
\end{example}

The minimum and characteristic polynomial operations have to return a variable for the polynomial. In the preceding examples, they returned X. However, the variable used depends on the matrix entries and you do not need to avoid X in the matrix. This point is illustrated in the following examples.

$\blacktriangleright$ Matrices + Minimum Polynomial

$\left(\vphantom{
\begin{array}{cc}
3X & x \\
5 & y
\end{array}
}\right.$$\begin{array}{cc}
3X & x \\
5 & y
\end{array}$$\left.\vphantom{
\begin{array}{cc}
3X & x \\
5 & y
\end{array}
}\right)$, minimum polynomial: -5x + 3Xy + $\left(\vphantom{ -3X-y}\right.$ -3X - y$\left.\vphantom{ -3X-y}\right)$λ + λ2

$\left(\vphantom{
\begin{array}{cc}
\lambda & x \\
X & y
\end{array}
}\right.$$\begin{array}{cc}
\lambda & x \\
X & y
\end{array}$$\left.\vphantom{
\begin{array}{cc}
\lambda & x \\
X & y
\end{array}
}\right)$, characteristic polynomial: Θ2 - $\left(\vphantom{ y+\lambda }\right.$y + λ$\left.\vphantom{ y+\lambda }\right)$Θ + λy - xX