We have the following operator
$\hat{H} = a|u_{1}\rangle \langle u_{2}| + a|u_{2}\rangle \langle u_{1}|$, with a = real constant and $|u_{i}\rangle $ an orthonormal base.
The matrix representation, H, of this operator in the $|u_{i}\rangle $ base is
\begin{pmatrix} 0 & a\\ a & 0 \end{pmatrix}
the eigenvalues are a and -a, and the corresponding orthonormal eigenvectors are $|\phi_{1}\rangle = \frac{1}{\sqrt2}$(1 1) and $|\phi_{2}\rangle = \frac{1}{\sqrt2}$(1 -1).
We want to find the matrix representation, H', of the operator $\hat{H}$ in the base $|\phi_{i}\rangle $.
So, I know we can do that multiplying matrixes like H' = SHS*, where S is the change of basis matrix with S$_{mn}$ = <u$_{m}$|$\phi_{n}$>. But our professor told us he didn't want us to do it that way, that there is a faster one, but I can't seem to find it. Any help would be appreciated.
By definition, $\hat{H}|\phi_1\rangle=a|\phi_1\rangle$ and $\hat{H}|\phi_2\rangle=-a|\phi_2\rangle$ thus the matrix of $\hat{H}$ in the basis $|\phi_i\rangle$ is simply $$ \begin{pmatrix} a & 0 \\ 0 &-a \end{pmatrix} $$