Answer to Question #106348 in Linear Algebra for jonash

Question

Let λ ∈ R be an eigenvalue of an orthogonal matrix A. Show that λ = ±1.
(Hint: consider the norm of Av, where v is an eigenvector of A associated with the
eigenvalue λ.)

Also, find diagonal orthogonal matrices B, C such that 1 is an eigenvalue of B
and −1 is an eigenvalue of C.

Answer 1

A square matrix $A$ is said to be orthogonal if $AA'=I$ $=A'A$

where $A'=$ Transpose of $A$ and $I=$ Identity matrix.

Suppose $\lambda \in R$ be a eigenvalue of $A$ .

Then there exists a non zero eigenvector $X$ such that

$AX=\lambda X$ $......(1)$

Taking transpose of both sides of the above equality, we get

$(AX)'=(\lambda X)'$

$\implies$ $X'A'=X' \lambda$

$\implies$ $X'A'=\lambda X'$

Multiplying both sides by $AX$ we get,

$X'A'AX=\lambda X'AX$

$\implies$$X'X=\lambda X' \lambda X$ [from equation (1)]

$\implies X'X={ \lambda }^2 X'X$

$\implies (1- { \lambda}^2)X'X=0$

Since ,$X \neq0 \implies X'X\neq0.$

Therefore,$(1-\lambda^2)=0$

$\implies \lambda^2=1$

Hence $\lambda =1,-1$ .

$(Proved)$ .

Let $B=$ $\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}$ which is a diagonal matrix and $BB'=B'B=I$

Hence ,$B$ is an othogonal matrix ,whose eigen values are $+1, +1.$

Let $C=\begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix}$ which is a diagonal matrix and $CC'=C'C=I$

Hence ,$C$ is an orthogonal matrix, whose eigen values are $-1,-1.$ .