Answer to Question #102641 in Linear Algebra for BIVEK SAH

Question #102641
Show that every eigenvalue of a unitary matrix is of unit modulus.
1
Expert's answer
2020-02-17T11:16:06-0500

Let UU be unitary matrix, λ\lambda be eigenvalue with corresponding eigenvector xx.

By the definition of unitary operator we have (x,x)=(Ux,Ux)(x,x)=(Ux,Ux) .

But Ux=λxUx=\lambda x , so (x,x)=(λx,λx)=λλ(x,x)=λ2(x,x)(x,x)=(\lambda x,\lambda x)=\lambda\overline{\lambda}(x,x)=|\lambda|^2(x,x) .

Since (x,x)0(x,x)\neq 0 , we have λ2=1|\lambda|^2=1 , that is λ=1|\lambda|=1 .


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