How to check whether a polynomial is annihilating polynomial?

Question

When I what to find the minimal polynomial of a matrix, first I want to get the characteristic polynomial of the matrix. But how can I check whether some polynomial is annihilating polynomial to be the minimal polynomial?

Answer 1

Let me explain with this little example:

Consider $A=\left(\begin{array}{cc}2&0\\0&2\end{array}\right)$ and $B=\left(\begin{array}{cc}2&1\\0&2\end{array}\right)$.

Both matrices have as their characteristic polynomials $(x-2)^2$, but the minimal polynomials change: we have $\mu_A(x)=x-2$ but $\mu_B(x)=(x-2)^2$.

Both matrices are into canonical form. Because of this is easy to say the minimals and the characteristics.

For other type of matrices, no necessarily in canonical form you can process them to pseudo-diagonalization with a robot (like wolfram alpha maybe), so the minimal polynomial should be crystal clear to infer.

Answer 2

The usual way to find the minimal polynomial is to factor the characteristic polynomial, since given a matrix $A$, $\mu_A\mid \chi_A$. If we have that $\chi_A = \prod_{i=1}^{k} (x - \lambda_i)^{p_i}$, then any one of the product combinations could be a minimal polynomial for $A$, so what you do is begin by trying small degree combinations, and building up towards the characteristic polynomial.

For example, if a matrix has characteristic polynomial $\chi_A(x) = (x - 1)(x - 3)^2$, then you could try $(x-1)$, $(x-3)$ and $(x-1)(x-3)$; if neither annihilates $A$, then $\chi_A$ will.