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. ${\displaystyle U\in \mathbb {C} ^{n\times n}}$ ${\displaystyle U^{H}}$ ${\displaystyle I}$

${\displaystyle U^{H}U=I}$

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 ${\displaystyle U}$ ${\displaystyle u_{1},\ldots ,u_{n}}$

${\displaystyle u_{i}^{H}\cdot u_{j}=\delta _{ij}={\begin{cases}1&{\text{falls}}~i=j\\0&{\text{sonst}}\end{cases}}}$

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. ${\displaystyle \delta _{ij}}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle U}$ ${\displaystyle U^{T}}$

Examples

The matrix

${\displaystyle U={\begin{pmatrix}0&i\\i&0\end{pmatrix}}}$

is unitary because it applies

${\displaystyle U^{H}\,U={\begin{pmatrix}0&-i\\-i&0\end{pmatrix}}\cdot {\begin{pmatrix}0&i\\i&0\end{pmatrix}}={\begin{pmatrix}-i^{2}&0\\0&-i^{2}\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I}$.

The matrix too

${\displaystyle U={\frac {1}{2}}{\begin{pmatrix}1+i&1-i\\1-i&1+i\end{pmatrix}}}$

is unitary because it applies

${\displaystyle U^{H}\,U={\frac {1}{2}}{\begin{pmatrix}1-i&1+i\\1+i&1-i\end{pmatrix}}\cdot {\frac {1}{2}}{\begin{pmatrix}1+i&1-i\\1-i&1+i\end{pmatrix}}={\frac {1}{4}}{\begin{pmatrix}2(1-i)(1+i)&(1-i)^{2}+(1+i)^{2}\\(1+i)^{2}+(1-i)^{2}&2(1+i)(1-i)\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}=I}$.

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 ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle U^{H}=U^{-1}}$.

The inverse of a matrix is precisely the matrix for which ${\displaystyle U}$ ${\displaystyle U^{-1}}$

${\displaystyle U\,U^{-1}=U^{-1}\,U=I}$

applies. The converse is also true, and every matrix whose adjoint is equal to its inverse is unitary, because then it is true ${\displaystyle U}$

${\displaystyle U^{H}\,U=U^{-1}\,U=I}$.

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

${\displaystyle UU^{H}=I}$.

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 ${\displaystyle x\in \mathbb {C} ^{n}}$ ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle \|U\,x\|_{2}=\|x\|_{2}}$.

Furthermore, the standard scalar product of two vectors is invariant with respect to the multiplication with a unitary matrix , i.e. ${\displaystyle x,y\in \mathbb {C} ^{n}}$ ${\displaystyle U}$

${\displaystyle \left\langle U\,x,U\,y\right\rangle =\left\langle x,y\right\rangle }$.

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

${\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ^{n},\quad x\mapsto U\,x}$

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. ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle \mathbb {C} ^{n}}$

Determinant

For the complex amount of the determinant of a unitary matrix we have ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle |\det U|=1}$,

which with the help of the determinants product set over

${\displaystyle \det U\cdot {\overline {\det U}}=\det U\cdot \det {\bar {U}}=\det U\cdot \det U^{H}=\det(UU^{H})=\det I=1}$

follows.

Eigenvalues

The eigenvalues ​​of a unitary matrix also all have the absolute value one, so they are of the form ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle \lambda =e^{it}}$

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 ${\displaystyle t\in \mathbb {R} }$ ${\displaystyle x}$ ${\displaystyle \lambda }$

${\displaystyle \|x\|_{2}=\|U\,x\|_{2}=\|\lambda \,x\|_{2}=|\lambda |\,\|x\|_{2}}$

and therefore . ${\displaystyle |\lambda |=1}$

Diagonalisability

A unitary matrix is normal , that is, it holds ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle U\,U^{H}=U^{H}\,U}$,

and therefore diagonalizable . After the spectral theorem, there is another unitary matrix such that ${\displaystyle V\in \mathbb {C} ^{n\times n}}$

${\displaystyle V^{-1}\,U\,V=D}$

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. ${\displaystyle D\in \mathbb {C} ^{n\times n}}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle U}$

Norms

The spectral norm of a unitary matrix is ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle \|U\|_{2}=\max _{\|x\|_{2}=1}\|U\,x\|_{2}=\max _{\|x\|_{2}=1}\|x\|_{2}=1}$.

For the Frobenius norm , the Frobenius scalar product applies accordingly

${\displaystyle \|U\|_{F}={\sqrt {\langle U,U\rangle _{F}}}={\sqrt {\langle I,I\rangle _{F}}}={\sqrt {n}}}$.

The product with a unitary matrix receives both the spectral norm and the Frobenius norm of a given matrix , because it holds ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle \|U\,A\|_{2}=\max _{\|x\|_{2}=1}\|U\,A\,x\|_{2}=\max _{\|x\|_{2}=1}\|A\,x\|_{2}=\|A\|_{2}}$

and

${\displaystyle \|U\,A\|_{F}={\sqrt {\langle U\,A,U\,A\rangle _{F}}}={\sqrt {\langle A,A\rangle _{F}}}=\|A\|_{F}}$.

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 ${\displaystyle U\in \mathbb {C} ^{n\times n}}$ ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle A\,A=A}$

${\displaystyle B=U\,A\,U^{H}}$

also idempotent, because

${\displaystyle B\,B=U\,A\,U^{H}U\,A\,U^{H}=U\,A\,A\,U^{H}=U\,A\,U^{H}=B}$.

Unitary matrices as a group

→ Main article : Unitary 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 ${\displaystyle \mathrm {GL} (n,\mathbb {C} )}$ ${\displaystyle I}$ ${\displaystyle \mathrm {U} (n)}$ ${\displaystyle U,V\in \mathbb {C} ^{n\times n}}$

${\displaystyle (U\,V)^{H}\,(U\,V)=V^{H}\,U^{H}\,U\,V=V^{H}\,V=I}$.

Furthermore, the inverse of a unitary matrix is also unitary because it holds ${\displaystyle U\in \mathbb {C} ^{n\times n}}$

${\displaystyle U^{-H}\,U^{-1}=U^{-H}\,U^{H}=(U\,U^{-1})^{H}=I^{H}=I}$.

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 . ${\displaystyle \mathrm {SU} (n)}$

use

Matrix decompositions

With the help of a singular value decomposition , each matrix can be converted into a product ${\displaystyle A\in \mathbb {C} ^{m\times n}}$

${\displaystyle A=U\,\Sigma \,V^{H}}$

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 . ${\displaystyle U\in \mathbb {C} ^{m\times m}}$ ${\displaystyle \Sigma \in \mathbb {C} ^{m\times n}}$ ${\displaystyle V\in \mathbb {C} ^{n\times n}}$ ${\displaystyle \Sigma }$ ${\displaystyle A}$

A square matrix can also be used as a product by means of the polar decomposition ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle A=U\,P}$

a unitary matrix and a positive semidefinite Hermitian matrix can be factored. ${\displaystyle U\in \mathbb {C} ^{n\times n}}$ ${\displaystyle P\in \mathbb {C} ^{n\times n}}$

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 ${\displaystyle (V,\langle \cdot ,\cdot \rangle )}$ ${\displaystyle n}$ ${\displaystyle f\colon V\to V}$ ${\displaystyle \{e_{1},\dotsc ,e_{n}\}}$ ${\displaystyle V}$

${\displaystyle A_{f}=(a_{ij})\in \mathbb {R} ^{n\times n}}$

represent, where for is. The mapping matrix is now unitary if and only if is a unitary mapping . This follows from ${\displaystyle f(e_{j})=a_{1j}e_{1}+\dotsb +a_{nj}e_{n}}$ ${\displaystyle j=1,\dotsc ,n}$ ${\displaystyle A_{f}}$ ${\displaystyle f}$

${\displaystyle \langle f(v),f(w)\rangle =(A_{f}x)^{H}(A_{f}y)=x^{H}A_{f}^{H}A_{f}y=x^{H}y=\langle v,w\rangle }$,

where and are. ${\displaystyle v=x_{1}e_{1}+\dotsb +x_{n}e_{n}}$ ${\displaystyle w=y_{1}e_{1}+\dotsb +y_{n}e_{n}}$

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).