Example of diagonalisable matrix with given property

Question

Let $A\in M_5(\mathbb C)$ satisfying $(A^2-I)^2=0$ and $A$ is not diagonal matrix.

Then I have To find matrix A

But I tried but adding some terms in up to diagonal Nilpotency occur Which prevent form diagonalisable.

Please Help me to find example

Answer 1

Write $(A-Id)^2(A+Id)^2$ instead of $(A^2-I)^2$. Consider for instance the $(4, 4)$ matrix with diagonal entries $1,1,-1,-1, 1$, the ligne just over the diagonal $1,0,1,0$ and ll other coefficients $0$.

You can also find diagonalisable matrix : any symmetry does the job.

Answer 2

This is an example: $$ A= \left[\begin{array}{cccc} -11 & 6 & 0 & 0 & 0\\ -20 & 11 & 0 & 0 & 0\\ 0 & 0 & -11 & 6 & 0\\ 0 & 0 & -20 & 11 & 0\\ 0 & 0 & 0 & 0 & 1\end{array}\right] $$

Answer 3

If you want that the matrix is diagonalizable, the minimal polynomial must have distinct roots, so the candidate for the minimal polynomial is $x^2-1$. The eigenvalues must be $1$ and $-1$ (there must be at least two eigenvalues, because otherwise the matrix would necessarily be diagonal).

One of them has multiplicity $3$ and the other one $2$. Let's find a $2\times2$ nondiagonal matrix having eigenvalues $1$ and $-1$, for instance \begin{bmatrix} 1 & 1 \\ -2 & -1 \end{bmatrix} Then your example is $$ \begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\ -2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & -1 \end{bmatrix} $$