# Unitary matrix

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

[itex]U^*U = UU^* = I_n\,[itex]

where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse which is equal to its conjugate transpose U*.

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

[itex]\langle Gx, Gy \rangle = \langle x, y \rangle[itex]

so also a unitary matrix U satisfies

[itex]\langle Ux, Uy \rangle = \langle x, y \rangle[itex]

for all complex vectors x and y, where <.,.> stands now for the standard inner product on Cn. If A is an n by n matrix then the following are all equivalent conditions:

1. A is unitary
2. A* is unitary
3. the columns of A form an orthonormal basis of Cn with respect to this inner product
4. the rows of A form an orthonormal basis of Cn with respect to this inner product
5. A is an isometry with respect to the norm from this inner product

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). The same is true for the determinant.

All unitary matrices are normal, and the spectral theorem therefore applies to them.

A unitary matrix is called special if its determinant is 1.

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