Answer to Question #194058 in Linear Algebra for Mpopo

Question

Find an expression for a square matrix A satisfying A 2 = In, where In is the n × n identity matrix

Answer 1

For any matrix A = I - 2 * v*v^T, where v is a unit vector of dimension n, and I is identity matrix

A² = A * A = (I - 2 v v^T) * (I - 2 v v^T) =

I - I * (2 v v^T) - (2 v v^T) * I + (2 v v^T) * (2 v v^T) =

I - 4 v v^T + 4 v v^T v v^T =

I - 4 v v^T + 4 v (v^T v) v^T =

I - 4 v v^T + 4 v I v^T =

I - 4 v v^T + 4 v v^T = I

quod erat demonstrandum.