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Unitary matrices are invertible.

Created over 8 years ago, updated 10 days ago

Fact. Every unitary matrix is invertible.

By definition, if $U$ is unitary then $U^* U = I$, which means $U^{-1} = U^*$. The inverse of a unitary matrix is its conjugate transpose, which always exists.

Furthermore:

  • $|\det(U)| = 1$ (the determinant has absolute value 1)
  • The eigenvalues of a unitary matrix all have absolute value 1
  • The product of unitary matrices is unitary

Example. For $U = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, we have $U^{-1} = U^* = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$.