I want to calculate the eigenvalues of a matrix given below.
$A=\begin{pmatrix}1&1&1&1 \\ 1 & i & -1 & -i \\ 1 & -1 & 1 & -1 \\ 1 & -i &-1 &i\end{pmatrix}$
We can calculate $\det{(A-tE)}$ by the Laplace expansion(or cofactor expansion), but that method takes too much time for me. Since this question is from an exam question, I need a quick(and preferably easy) method to calculate the eigenvalues.
Of course, I have seen these questions:
Eigenvalues for $4\times 4$ matrix
Quick method for finding eigenvalues and eigenvectors in a symmetric $5 \times 5$ matrix?
However, the answers there do not seem to be applicable to our matrix. Is there any method that can be applied to our matrix?
Best Answer
The eigenvalues of $A^2=\pmatrix{4\\ &&&4\\ &&4\\ &4}$ are $4,4,4,-4$. Therefore three of the eigenvalues of $A$ are $\pm2$ and the remaining one is $\pm2i$. Since the trace of $A$ is $2+2i$, the four eigenvalues must be $2,2,-2,2i$.