TITLE: A matrix is unitary if, and only if, diagonalizable and all eigenvalues are on the unit circle
QUESTION [5 upvotes]: I would like to know if the class of unitary matrices is the same as the class of diagonalizable matrices who have all their eigenvalues on the unit circle.

REPLY [7 votes]: Every unitary matrix is diagonalizable and the eigenvalues are on the unit circle. However, the converse is not true, since unitary matrices have orthogonal eigenspaces, whereas your condition does not imply this.
As an example, $\begin{bmatrix} 1 & 1 \\ 0 & -1 \end{bmatrix}$ has the two different eigenvalues $\pm 1$, so it is diagonalizable with eigenvalues on the unit circle. However, it is not unitary.