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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 lying on the unit circle.
A square matrix is a unitary matrix if
where denotes the conjugate transpose and is the matrix inverse. For example,
is a unitary matrix.
Unitary matrices leave the length of a complex vector unchanged.
For real matrices, unitary is the same as orthogonal. In fact, there are some similarities between orthogonal matrices and unitary matrices. The rows of a
unitary matrix are a unitary basis. That is, each row has length one, and their Hermitian inner product is zero. Similarly, the columns are also a unitary basis. In fact, given any unitary basis, the matrix whose rows are that basis is a unitary matrix. It is automatically the case that the columns are another unitary basis.
The definition of a unitary matrix guarantees that
where is the identity matrix. In particular, a unitary matrix is always invertible, and . Note that transpose is a much simpler computation than inverse. A similarity transformation of a Hermitian matrix with a unitary matrix gives
Unitary matrices are normal matrices. If is a unitary matrix, then the permanen
The unitary matrices are precisely those matrices which preserve the Hermitian inner product
Also, the norm of the determinant of is . Unlike the orthogonal matrices, the unitary matrices are connected. If then is a special unitary matrix.
The product of two unitary matrices is another unitary matrix. The inverse of a unitary matrix is another unitary matrix, and identity matrices are unitary. Hence the set of unitary matrices form a group, called the unitary group.
Properties Of unitary matrix
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.
- 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).
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In mathematics, the unitary group of degree n, denoted U(n), is the group of n×n unitary matrices, with the group operation that of matrix multiplication. The unitary group is a subgroup of the general linear group GL (n, C).
In the simple case n = 1, the group U(1) corresponds to the circle group, consisting of all complex numbers with absolute value 1 under multiplication. All the unitary groups contain copies of this group.
The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of complex n×n skew-Hermitian matrices, with the Lie bracket given by the commutator.
The general unitary group (also called the group of unitary similitude) consists of all matrices A such that A * A is a nonzero multiple of the identity matrix, and is just the product of the unitary group with the group of all positive multiples of the identity matrix.
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