Diagonalizable Matrices with Eigenvalues of + or -1: A Proof

[trojansc82]

Homework Statement

Prove that if the eigenvalues of a diagonalizable matrix are all + or -1, then the matrix is equal to its inverse.

i) Let D = P^-1AP, where D is a diagonal matrix with + or -1 along its main diagonal.

ii) Find A in terms of P, P^-1, and D.

iii) Use the fact that D is the diagonal and the properties of the inverse of a product of matrices to expand to find A^-1.

iv) Conclude that A^-1 = A.

Homework Equations

The Attempt at a Solution

D * P^-1 = P^-1 AP *P^-1

P * D * P^-1 = P * P^-1 A

PDP^-1 = A

Not sure if I'm heading in the right direction. I am drawing a blank here.

 

[HallsofIvy]

Yes, $PDP^{-1}= A$. Further more if D is invertible (and it clearly is since it does not have a 0 on its diagonal), so is A and $A^{-1}= (PDP^{-1})^{-1}$

Now use the fact that $(ABC)^{-1}= C^{-1}B^{-1}A^{-1}$.

 

[trojansc82]

Ok, how do I show that D^-1 = D ?

 

[HallsofIvy]

What is the inverse of
$$\begin{bmatrix}a & 0 \\ 0 & b\end{bmatrix}$$

What is the inverse of
$$\begin{bmatrix}a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c\end{bmatrix}$$

 

[trojansc82]

1. 1/ab * $$\begin{bmatrix}b & 0 \\ 0 & a\end{bmatrix}$$

2. 1/abc * $$\begin{bmatrix}c & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & a\end{bmatrix}$$