Unitary matrix
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In mathematics, a unitary matrix is a n by n complex matrix U satisfying the condition
- <math>U^*U = UU^* = I_n\,<math>
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,
- <math>\langle Gx, Gy \rangle = \langle x, y \rangle<math>
so also a unitary matrix U satisfies
- <math>\langle Ux, Uy \rangle = \langle x, y \rangle<math>
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:
- A is unitary
- A* is unitary
- the columns of A form an orthonormal basis of Cn with respect to this inner product
- the rows of A form an orthonormal basis of Cn with respect to this inner product
- 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.