Unitary matrix

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

$U^{\dagger} U = UU^{\dagger} = I_n\,$

where $I_n\,$ 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 $U^{\dagger} \,$

$U^{-1} = U^{\dagger} \,\;$

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,

$\langle Gx, Gy \rangle = \langle x, y \rangle$

so also a unitary matrix $U$ satisfies

$\langle Ux, Uy \rangle = \langle x, y \rangle$

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:

$U \,$ is unitary
$U^{\dagger} \,$ is unitary
the columns of $U \,$ form an orthonormal basis of $\mathbb{C}^n$ with respect to this inner product
the rows of $U \,$ form an orthonormal basis of $\mathbb{C}^n$ with respect to this inner product
$U \,$ is an isometry with respect to the norm from this inner product
U is a normal matrix with eigenvalues contained in the unit circle.

[edit] Properties of unitary matrices

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

$U = V\Sigma V^{\dagger}\;$

where

V

is unitary, and

Σ

is diagonal and unitary. That is, a unitary matrix is diagonalizable by a unitary matrix.

For any unitary matrix $U$ , the following hold:

$U$ is invertible.
$U^{-1}=U^{\dagger}.$
$| det(U) | = 1.$
$U^{\dagger}$ is unitary.
$U$ preserves length $\|Ux\|_2=\|x\|_2.$
$U$ has complex eigenvalues of modulus 1. ^[1]
It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e., they lie on the unit circle centered at 0 in the complex plane).
For any n, the set of all n by n unitary matrices with matrix multiplication forms a group.
Any 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 (depending on $M$ , of course).^[2]