Let $A$ be a unitary matrix over field $F$ ($F=\mathbb{R},\mathbb{C}$) Prove that if all its eigenvalues equal to $1$ then $A$ must be the identity matrix $I$.
I am having a hard time figuring out where to begin. Thanks.
2026-04-26 04:22:47.1777177367
Unitary matrix with all its eigenvalues equal to 1 must be the identity matrix
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If the matrix is unitary, then it is diagonalisable, and if all eigenvalues are one, then there exists a matrix $V$ such that:
$$A=VIV^{-1}=VV^{-1}=I$$
Being $A$ our matrix.
EDIT:
As I saw in the comments, your actual question is how do we know that unitary matrices are diagonalizable. Here you have the proof:
http://www.math.tamu.edu/~dallen/m640_03c/lectures/chapter4.pdf
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