# Reference for matrices with all eigenvalues 1 or -1

MathOverflow Asked on December 7, 2020

In a homological algebra problem I am in the situation that I have an invertible (over $$mathbb{Z}$$) integer matrix $$X$$ and a permutation matrix $$Y$$ such that $$N:=XY$$ is a matrix with all eigenvalues equal to 1 or -1.

Question 1 : Does such a situation appear already in other situations/fields in algebra/combinatorics? Is there an interpretation for this or does it have a combinatorial meaning?

Question 2: Do (invertible integer) matrices with all eigenvalues equal to 1 or -1 have a name, or do matrices $$X$$ as above have a name? Are they studied in the literature?

If $$N$$ is diagonalizable then $$N=Q Lambda Q^{−1,}$$ hence $$N^2=Q Lambda Q^{−1} Q Lambda Q^{−1}=Q Lambda^2 Q^{−1}$$

Since this $$Lambda$$ is diagonal with $$pm 1$$ on its diagonal, $$Lambda^2=I_n$$

hence $$N^2=QQ^{−1}=I_n$$

These don't have (as far as I know) "names" but the properties may help you along the way.

Answered by Kevin on December 7, 2020