Question #160253

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

True or false with full explanation


1
Expert's answer
2021-02-24T07:07:16-0500

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 UU=IU U † = I  Thus, if

Uv=λvU | v ⟩ = λ | v ⟩  (1)

then also

vU=vλ.⟨ v | U † = ⟨ v | λ ∗ . (2)

Combining (1) and (2) leads to

vv=vUUv=vλλv=λ2vv⟨ v | v ⟩ = ⟨ v | U † U | v ⟩ = ⟨ v | λ ∗ λ | v ⟩ = | λ | 2 ⟨ v | v ⟩

Assuming λ0λ ≠ 0, we thus have

λ2=1.| λ | ^2 = 1 .

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


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