Find a matrix $X$ given $X^4$ Find the matrix $X$ such that
$$X^4=\begin{bmatrix}
3&0&0\\
0&3&1\\
0&0&0
\end{bmatrix}$$
This problem I can't work,and I think let the matrix the eigenvalue is $\lambda$,then
$\lambda^4$ is $$\begin{bmatrix}
3&0&0\\
0&3&1\\
0&0&0
\end{bmatrix}$$
eigenvalue?Thank you for your help.

You can look for a solution of the same form as $X$: calculating $$\begin{pmatrix} a & 0 & 0 \\ 0 & a & b \\ 0 & 0 & 0\end{pmatrix} \begin{pmatrix} a & 0 & 0 \\ 0 & a & b \\ 0 & 0 & 0\end{pmatrix} = a \cdot \begin{pmatrix} a & 0 & 0 \\ 0 & a & b \\ 0 & 0 & 0\end{pmatrix}$$ shows that $$\begin{pmatrix} a & 0 & 0 \\ 0 & a & b \\ 0 & 0 & 0\end{pmatrix}^4 = a^3 \begin{pmatrix} a & 0 & 0 \\ 0 & a & b \\ 0 & 0 & 0\end{pmatrix}.$$ In particular, you can take $a = 3^{1/4}$ and $b = 3^{-3/4}$ for one possibility of $X$.
