Answer to Question #160253 in Linear Algebra for Sarita bartwal

Question

If A is a unitary matrix, then all its eigen values are 1.

True or false with full explanation

Answer 1

If A is a unitary matrix, then all its Eigen values are 1.

this statement is true because:

The eigenvalues and eigenvectors of unitary matrices have some special properties.

If U is unitary, then $U U † = I$  Thus, if

$U | v ⟩ = λ | v ⟩$  (1)

then also

$⟨ v | U † = ⟨ v | λ ∗ .$ (2)

Combining (1) and (2) leads to

$⟨ v | v ⟩ = ⟨ v | U † U | v ⟩ = ⟨ v | λ ∗ λ | v ⟩ = | λ | 2 ⟨ v | v ⟩$

Assuming $λ ≠ 0$, we thus have

$| λ | ^2 = 1 .$

Thus, the eigenvalues of a unitary matrix are unimodular, that is, they have norm 1, and hence can be written as $e^{i\alpha}$ for some $\alpha$