# Unitary matrix

 related topics {math, number, function}

In mathematics, a unitary matrix is an $n\times n$ complex matrix U satisfying the condition

where In is the identity matrix in n dimensions and $U^{\dagger}$ is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if and only if it has an inverse which is equal to its conjugate transpose $U^{\dagger} \,$

A unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors,

so also a unitary matrix U satisfies

for all complex vectors x and y, where $\langle\cdot,\cdot\rangle$ stands now for the standard inner product on $\mathbb{C}^n$.

If $U \,$ is an n by n matrix then the following are all equivalent conditions:

## Contents

### Properties

• All unitary matrices are normal, and the spectral theorem therefore applies to them. Thus every unitary matrix U has a decomposition of the form

For any unitary matrix U, the following hold:

• $U \,$ is invertible, with $U^{-1}=U^{\dagger}$.
• $U^{\dagger}$ is also unitary.
• $U \,$ preserves length ("isometry"): $\|Ux\|_2=\|x\|_2$.
• $U \,$ has complex eigenvalues of modulus 1. [1]
• For any n, the set of all n by n unitary matrices with matrix multiplication forms a group, called U(n).
• Any unit-norm matrix is the average of two unitary matrices. As a consequence, every $n \times n$ matrix M is a linear combination of two unitary matrices.[2]