In mathematics, a unitary matrix is an complex matrix U satisfying the condition
where In is the identity matrix in n dimensions and 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
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 stands now for the standard inner product on .
If is an n by n matrix then the following are all equivalent conditions:
- 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:
- is invertible, with .
- is also unitary.
- preserves length ("isometry"): .
- has complex eigenvalues of modulus 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 matrix M is a linear combination of two unitary matrices.
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