Unitary matrix

Definition

Let $n \ge 1$ and let $U$ be a complex $n \times n$ matrix. The matrix $U$ is unitary when $$U^* U = I_n,$$ where $U^*$ is the conjugate transpose and $I_n$ is the identity matrix.

Remarks

Equivalently, $U$ is invertible with $U^{-1} = U^*$, and then also $U U^* = I_n$. Equivalently, the columns of $U$ form an orthonormal basis of $\C^n$ for the standard inner product. The eigenvalues of a unitary matrix have absolute value $1$, and $\abs{\det U} = 1$. A real unitary matrix is an orthogonal matrix.

Explanations

L1 human

Unitary matrices are the complex version of orthogonal matrices. They keep all lengths and inner products in $\C^n$. In quantum mechanics, they describe how a closed system evolves in time.