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 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:
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- U is a normal matrix with eigenvalues contained in the unit circle.
Contents |
[edit] Properties of unitary matrices
- All unitary matrices are normal, and the spectral theorem therefore applies to them. Thus every unitary matrix U has a decomposition of the form
- where V is unitary, and Σ is diagonal and unitary. That is, a unitary matrix is diagonalizable by a unitary matrix.
For any unitary matrix U, the following hold:
- U is invertible.

- | det(U) | = 1.
is unitary.- U preserves length

- U has complex eigenvalues of modulus 1. [1]
- 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).
- For any n, the set of all n by n unitary matrices with matrix multiplication forms a group.
- Any matrix is the average of two unitary matrices. As a consequence, every
matrix M is a linear combination of two unitary matrices (depending on M, of course).[2]
[edit] See also
- Hermitian matrix
- symplectic matrix
- unitary group
- special unitary group
- unitary operator
- defect of a unitary matrix
- matrix decomposition






