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Unitary matrix
From Wikipedia, the free encyclopedia
In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition
where 
is the identity matrix 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 the same thing as 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 Cn. If 
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 Cn with respect to this inner product
- the rows of form an orthonormal basis of Cn with respect to this inner product
- 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. Thus every unitary matrix U has a decomposition of the form
- U = VΣV *
where V is unitary, and Σ is diagonal and unitary.
For any n, the set of all n by n unitary matrices with matrix multiplication form a group.
A unitary matrix is called special if its determinant is 1.