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Unitary matrix


Unitary matrix

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:

  1. is unitary
  2. is unitary
  3. the columns of form an orthonormal basis of with respect to this inner product
  4. the rows of form an orthonormal basis of with respect to this inner product
  5. is an isometry with respect to the norm from this inner product
  6. 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).

Unitary group

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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