# Unitary matrix

In linear algebra, a **unitary matrix** is a complex square matrix whose row and column vectors are orthonormal with respect to the standard scalar product. Thus the inverse of a unitary matrix is also its adjoint .

By multiplying by a unitary matrix, both the Euclidean norm and the standard scalar product of two vectors are preserved. Every unitary mapping between two finite-dimensional scalar product spaces can be represented by a unitary matrix, depending on the choice of an orthonormal basis . The set of unitary matrices of a fixed size forms the unitary group with the matrix multiplication as a link .

Unitary matrices are used in singular value decomposition , discrete Fourier transformation and in quantum mechanics , among other things . A real unitary matrix is called an orthogonal matrix .

## definition

A complex square matrix is called unitary if the product with its adjoint matrix results in the unit matrix , i.e.

applies. Are the column vectors of the matrix with designated, this condition is equivalent to saying that there is always the standard scalar of two column vectors

results, where the Kronecker delta is. The column vectors of a unitary matrix thus form an orthonormal basis of the coordinate space . This also applies to the row vectors of a unitary matrix, because with the transposed matrix is also unitary.

## Examples

The matrix

is unitary because it applies

- .

The matrix too

is unitary because it applies

- .

In general, every orthogonal matrix is unitary, because for matrices with real entries the adjoint corresponds to the transpose.

## properties

### Inverse

A unitary matrix is always regular due to the linear independence of its row and column vectors . The inverse of a unitary matrix is equal to its adjoint, that is, it applies

- .

The inverse of a matrix is precisely the matrix for which

applies. The converse is also true, and every matrix whose adjoint is equal to its inverse is unitary, because then it is true

- .

In addition, the adjoint of a unitary matrix is also unitary, because

- .

### Invariance of norm and scalar product

If a vector is multiplied by a unitary matrix , the Euclidean norm of the vector does not change, that is

- .

Furthermore, the standard scalar product of two vectors is invariant with respect to the multiplication with a unitary matrix , i.e.

- .

Both properties follow directly from the displacement property of the standard scalar product. Hence the figure represents

represents a congruence map in unitary space . Conversely, the mapping matrix is unitary with respect to the standard basis of any linear map im that receives the standard scalar product. Because of the polarization formula , this also applies to the mapping matrix of every linear mapping that receives the Euclidean norm.

### Determinant

For the complex amount of the determinant of a unitary matrix we have

- ,

which with the help of the determinants product set over

follows.

### Eigenvalues

The eigenvalues of a unitary matrix also all have the absolute value one, so they are of the form

with . If there is an eigenvector that belongs to it, then due to the invariance with regard to the Euclidean norm and the absolute homogeneity of a norm , the following applies

and therefore .

### Diagonalisability

A unitary matrix is normal , that is, it holds

- ,

and therefore diagonalizable . After the spectral theorem, there is another unitary matrix such that

holds, where is a diagonal matrix with the eigenvalues of . The column vectors of are then pairwise orthonormal eigenvectors of . Thus the eigenspaces of a unitary matrix are also orthogonal in pairs.

### Norms

The spectral norm of a unitary matrix is

- .

For the Frobenius norm , the Frobenius scalar product applies accordingly

- .

The product with a unitary matrix receives both the spectral norm and the Frobenius norm of a given matrix , because it holds

and

- .

This means that the condition of a matrix with regard to these norms is retained after multiplication with a unitary matrix.

### Preservation of idempotence

If there is a unitary and an idempotent matrix , therefore , then the matrix is

also idempotent, because

- .

## Unitary matrices as a group

The set of regular matrices of fixed size and the matrix multiplication as a link form a group , the general linear group . The identity matrix serves as a neutral element . The unitary matrices form a subgroup of the general linear group, the unitary group . The product of two unitary matrices is unitary again because it holds

- .

Furthermore, the inverse of a unitary matrix is also unitary because it holds

- .

The unitary matrices with determinant one in turn form a subgroup of the unitary group, the special unitary group . The unitary matrices with determinant minus one do not form a subgroup of the unitary group, because they lack the neutral element, only a secondary class .

## use

### Matrix decompositions

With the help of a singular value decomposition , each matrix can be converted into a product

a unitary matrix , a diagonal matrix and the adjoints of another unitary matrix . The diagonal entries of the matrix are then the singular values of .

A square matrix can also be used as a product by means of the polar decomposition

a unitary matrix and a positive semidefinite Hermitian matrix can be factored.

### Unitary mappings

If a -dimensional complex scalar product space , then every linear mapping after choosing an orthonormal basis for can be determined by the mapping matrix

represent, where for is. The mapping matrix is now unitary if and only if is a unitary mapping . This follows from

- ,

where and are.

### Physical applications

Unitary matrices are also often used in quantum mechanics within the framework of matrix mechanics . Examples are:

- the Dirac matrices
- the Pauli matrices
- the S matrix
- the CKM matrix
- the PMNS matrix

Another important application of unitary matrices is the discrete Fourier transform of complex signals.

## literature

- Gerd Fischer : Linear Algebra. (An introduction for first-year students) . 14th revised edition. Vieweg, 2003, ISBN 3-528-03217-0 .
- Jörg Liesen, Volker Mehrmann : Linear Algebra . Springer, 2011, ISBN 978-3-8348-8290-5 .
- Eberhard Zeidler , Wolfgang Hackbusch (Hrsg.): Pocket book of mathematics . tape 1 . Springer, 2012, ISBN 978-3-8351-0123-4 .

## Web links

- Oxana A. Ivanova: Unitary matrix . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).
- Todd Rowland:
*Unitary Matrix*. In:*MathWorld*(English).