Doubts

Question

Eigenvalues of an Idempotentmatrix matrix A is  0 or 1 right or wrong?

Answer 1

let $A$ be an idempotent matrix $(A = A^2 )$

let $\lambda$ be an eigenvalue of A and let $X$ be an eigenvector corresponding to the eigenvalue $\lambda$.

$AX = \lambda X $

$A^2X = \lambda^2 X $                                        $(A = A^2)$

from above we can say,

$AX = A^2X$ and $\lambda X = \lambda^2X$

$\lambda = \lambda ^2$

the above is satisfied only when $\lambda = \text{0 or 1}$

An idempotent matrix is always diagonalizable and its eigenvalues are either 0 or 1