# Unitary matrix

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where

*I*_{n}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 transposeA unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix

*G*preserves the (real) inner productof two real vectors,so also a unitary matrix

*U*satisfiesIf is an

*n*by*n*matrix then the following are all equivalent conditions:- is unitary
- is unitary
- the columns of form an orthonormal basis of with respect to this inner product
- the rows of form an orthonormal basis of with respect to this inner product
- is an isometry with respect to the norm from this inner product
- is a normal matrix with eigenvalues lying on the unit circle.

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