My question has a couple of parts.

First off, I'm interested in finding ALL possible n x n Hermitian matrices for a given n > 2.

Secondly, I'd like to find those matrices whose eigenvalues are $\pm 1$.

I feel there must be some sort of generic parametrization technique kind of like that for unitary matrices defined in terms of $\sin\theta$ and $\cos\theta$, but I'm not being able to find any.

This is what I've reasoned out so far for n=3 (I apologise for how naive and clumsy it seems):

A generic Hermitian matrix takes the form
H=\begin{array}{ccc} A & B & C \\ B* & D & E \\ C* & E* & F \end{array}

where A, D and F are real numbers. With the restriction of eigenvalues being $\pm 1$, we have the relations $det(H\pm I)=0$,

which gives us only two equations in 9 unknowns (as $B=b_1+ib_2$,$C=c_1+ic_2$, $E=e_1+ie_2$).

I've tried doing this by brute force by doing things like assuming A=1, E=B=1+i, and assigning other values as well to reduce the number of unknowns to 2, but this is definitely not the most efficient way, and I'm bound to make mistakes or miss out on some matrices.

Could someone please point me to a better method? And perhaps a parametrized form of Hermitian matrices, for 3 x 3 at least, if it exists?

Thanks in advance! :)


1 Answer 1


The best approach to finding Hermitian matrices with certain eigenvalues is to note that a matrix with real eigenvalues is Hermitian if and only if matrix that matrix is unitarily diagonalizable. That is, every Hermitian matrix $A$ can be written in the form $$ A= UDU^\dagger $$ where $U$ is unitary and $D$ is the diagonal matrix of eigenvalues.


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