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?

One Answer

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

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