Show that any eigenvalue of a unitary matrix U satisfies 1

Show that any eigenvalue of a unitary matrix U satisfies || = 1.

Solution

A unitary matrix is a square n-by-n matrix, U, satisfying U* U = I where U* represents the conjugate transpose of U and I is the multiplicative identity for n-by-n matrices.

is an eigenvalues of A ( unitary matrix)

Av = v

A*v = (1/)v ( By matrix property of determinant)

Thus , = 1/

^2 =1

= +/- 1

|| = 1. This means that the absolute value of any eigenvalue of a unitary matrix is one