Mathematics: A4x4 matrix a with real entries and a4 = i, what are the possible characteristic polynomials of a

A4x4 matrix a with real entries and a4 = i, what

Ответ:

eddsworldfrantic

13.03.2022

If $\mathbf A^4=\mathbf I$ , then the characteristic polynomial of $\mathbf A^4$ is

$p_{\mathbf A^4}(\lambda)=\det(\lambda\mathbf I-\mathbf A^4)=\det((\lambda-1)\mathbf I)=(\lambda-1)^4$

which means $\mathbf A^4$ has eigenvalues $\lambda=1$ .

We know that if $\chi$ is an eigenvalue of $\mathbf X$ , then $\chi^n$ is an eigenvalues of $\mathbf X^n$ .

So if $\lambda=1$ is the only eigenvalue of $\mathbf A^4$ , we know that $\pm\sqrt[4]\lambda=\pm1$ are the only possible eigenvalues of $\mathbf A$ . We can construct five possible characteristic polynomials for $\mathbf A$ in that case.

$p_{\mathbf A}(\lambda)=(\lambda-1)^4$
$p_{\mathbf A}(\lambda)=(\lambda+1)(\lambda-1)^3$
$p_{\mathbf A}(\lambda)=(\lambda+1)^2(\lambda-1)^2$
$p_{\mathbf A}(\lambda)=(\lambda+1)^3(\lambda-1)$
$p_{\mathbf A}(\lambda)=(\lambda+1)^4$