Conjugated matrix

The conjugated matrix , conjugated for short , is the matrix in mathematics that is created by complex conjugation of all elements of a given complex matrix. The conversion of a matrix into its conjugate matrix is ​​called conjugation of the matrix. The conjugation mapping that assigns its conjugates to a matrix is ​​always bijective , linear and self-inverse . Many parameters of conjugate matrices, such as trace , determinants and eigenvalues , are precisely the complex conjugates of the respective parameters of the output matrices .

The conjugate matrix is ​​used, for example, in the definition of the adjoint matrix , which results from conjugation and transposition of a given matrix. In addition, the conjugate matrix is ​​also used in the definition of the conjugate similarity of matrices.

definition

Is a complex matrix , ${\displaystyle A=(a_{ij})\in \mathbb {C} ^{m\times n}}$

${\displaystyle A={\begin{pmatrix}a_{11}&\dots &a_{1n}\\\vdots &&\vdots \\a_{m1}&\dots &a_{mn}\end{pmatrix}}}$

then the corresponding conjugate matrix is defined as ${\displaystyle {\bar {A}}\in \mathbb {C} ^{m\times n}}$

${\displaystyle {\bar {A}}=({\bar {a}}_{ij})={\begin{pmatrix}{\bar {a}}_{11}&\dots &{\bar {a}}_{1n}\\\vdots &&\vdots \\{\bar {a}}_{m1}&\dots &{\bar {a}}_{mn}\end{pmatrix}}}$.

The conjugate matrix results from the complex conjugation of all entries in the output matrix . Occasionally the conjugate matrix is ​​also noted by, but then there is a risk of confusion with the adjoint matrix , which is also designated. ${\displaystyle {\bar {A}}}$ ${\displaystyle A}$ ${\displaystyle A^{\ast }}$

Examples

The conjugate of the matrix

${\displaystyle A={\begin{pmatrix}1&2+i&3-2i\\4i&-5&-6-3i\end{pmatrix}}\in \mathbb {C} ^{2\times 3}}$

is the matrix

${\displaystyle {\bar {A}}={\begin{pmatrix}1&2-i&3+2i\\-4i&-5&-6+3i\end{pmatrix}}\in \mathbb {C} ^{2\times 3}}$.

For a complex matrix with only real entries, the conjugate is the same as the initial matrix .

properties

Calculation rules

The following calculation rules for conjugated matrices follow directly from the calculation rules for complex conjugation . It apply

${\displaystyle {\overline {z\cdot A}}={\bar {z}}\cdot {\bar {A}}}$

${\displaystyle {\overline {A+B}}={\bar {A}}+{\bar {B}}}$

${\displaystyle {\overline {A\cdot C}}={\bar {A}}\cdot {\bar {C}}}$

${\displaystyle {\bar {\bar {A}}}=A}$

for all matrices , and all scalars . ${\displaystyle A,B\in \mathbb {C} ^{m\times n}}$ ${\displaystyle C\in \mathbb {C} ^{n\times k}}$ ${\displaystyle z\in \mathbb {C} }$

Transposed

The conjugate of the transposed matrix is equal to the transpose of the conjugated matrix, that is

${\displaystyle {\overline {A^{T}}}=\left({\bar {A}}\right)^{T}}$.

This matrix is ​​called the adjoint matrix of and is usually referred to as or . ${\displaystyle A}$ ${\displaystyle A^{H}}$ ${\displaystyle A^{*}}$

Inverse

The conjugate of a regular matrix is always also regular. For the conjugate of the inverse of a regular matrix applies ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle {\overline {A^{-1}}}=\left({\bar {A}}\right)^{-1}}$.

The conjugate of the inverse matrix is ​​therefore equal to the inverse of the conjugate matrix.

Exponential and logarithm

The following applies to the matrix exponential of the conjugate of a square matrix ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle \exp({\bar {A}})={\overline {\exp A}}}$.

Correspondingly, the conjugate of a regular complex matrix applies to the matrix logarithm

${\displaystyle \ln({\bar {A}})={\overline {\ln A}}}$.

Conjugation mapping

The figure

${\displaystyle \mathbb {C} ^{m\times n}\to \mathbb {C} ^{m\times n},\quad A\mapsto {\bar {A}}}$,

which assigns its conjugate to a matrix is called conjugation mapping. Due to the above regularities, the conjugation mapping has the following properties.

• The conjugation mapping is always bijective , linear and self-inverse .
• In the matrix space , the conjugation mapping represents an automorphism . ${\displaystyle \mathbb {C} ^{m\times n}}$
• In the general linear group and in the matrix ring , the conjugation mapping (for ) also represents an automorphism. ${\displaystyle \operatorname {GL} (n,\mathbb {C} )}$ ${\displaystyle \mathbb {C} ^{n\times n}}$ ${\displaystyle m=n}$

Parameters

For the rank of the conjugates of a matrix we have ${\displaystyle A\in \mathbb {C} ^{m\times n}}$

${\displaystyle \operatorname {rang} {\bar {A}}=\operatorname {rang} (A)}$.

For the trace of the conjugates of a square matrix , however, the following applies ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle \operatorname {spur} {\bar {A}}={\overline {\operatorname {spur} (A)}}}$.

The same applies to the determinant of the conjugate of a square matrix

${\displaystyle \det {\bar {A}}={\overline {\det(A)}}}$.

For the characteristic polynomial of this results ${\displaystyle {\bar {A}}}$

${\displaystyle \chi _{\bar {A}}(\lambda )=\det(\lambda I-{\bar {A}})=\det {\overline {({\bar {\lambda }}I-A)}}={\overline {\det({\bar {\lambda }}I-A)}}={\overline {\chi _{A}({\bar {\lambda }})}}}$.

The eigenvalues of are therefore precisely the complex conjugates of the eigenvalues ​​of . The associated eigenvectors can also be chosen to be complex conjugate. ${\displaystyle {\bar {A}}}$ ${\displaystyle A}$

Norms

The following applies to the Frobenius norm and the spectral norm of the conjugates of a matrix ${\displaystyle A\in \mathbb {C} ^{m\times n}}$

${\displaystyle \|{\bar {A}}\|_{F}=\|A\|_{F}}$   and   . ${\displaystyle \|{\bar {A}}\|_{2}=\|A\|_{2}}$

This also applies to the row sum norm and the column sum norm of the conjugates

${\displaystyle \|{\bar {A}}\|_{1}=\|A\|_{1}}$   and   . ${\displaystyle \|{\bar {A}}\|_{\infty }=\|A\|_{\infty }}$

These matrix norms are therefore retained under conjugation.

use

Special matrices

The conjugate matrix is ​​used in linear algebra for the following definitions, among others:

• The adjoint matrix is that the matrix, which is formed by conjugation and transposition of a given complex matrix, ie . ${\displaystyle A^{H}={\overline {A^{T}}}=({\bar {A}})^{T}}$
• A Hermitian matrix is a complex square matrix whose transpose is equal to its conjugate, that is . ${\displaystyle A^{T}={\bar {A}}}$
• A skewed Hermitian matrix is a complex square matrix whose transpose is equal to the negative of its conjugate, that is . ${\displaystyle A^{T}=-{\bar {A}}}$
• A complex matrix is real if and only if it is equal to its conjugate matrix, that is, if holds. ${\displaystyle A={\bar {A}}}$

Product with the conjugate

For a complex number , the number as the square of the absolute value is always real and nonnegative. For a complex square matrix , however, the matrix need not necessarily be real. The determinant of is, however, always real and nonnegative, because it applies with the determinant product theorem ${\displaystyle z}$ ${\displaystyle z{\bar {z}}}$ ${\displaystyle A\in \mathbb {C} ^{n\times n}}$ ${\displaystyle A{\bar {A}}}$ ${\displaystyle A{\bar {A}}}$

${\displaystyle \det(A{\bar {A}})=\det(A)\cdot \det({\bar {A}})=\det(A)\cdot {\overline {\det(A)}}}$.

The eigenvalues ​​of the matrix do not all have to be real either, but the non-real eigenvalues ​​appear in complex conjugate pairs. The matrix occurs, for example, when analyzing complex symmetric matrices . ${\displaystyle A{\bar {A}}}$ ${\displaystyle A{\bar {A}}}$

Conjugated similarity

Two square matrices are called similarly conjugate ( English consimilar ) if a regular matrix exists such that ${\displaystyle A,B\in \mathbb {C} ^{n\times n}}$ ${\displaystyle S\in \mathbb {C} ^{n\times n}}$

${\displaystyle B=S^{-1}~A~{\bar {S}}}$

applies. The conjugate similarity, like the normal similarity, represents an equivalence relation on the set of square matrices. Two regular matrices are conjugate to one another, if and only if the matrix is similar to the matrix . ${\displaystyle A,B\in \mathbb {C} ^{n\times n}}$ ${\displaystyle A{\bar {A}}}$ ${\displaystyle B{\bar {B}}}$

literature

• Siegfried Bosch : Linear Algebra . Springer, 2006, ISBN 3-540-29884-3 .
• Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, ISBN 0-521-46713-6 .

Individual evidence

1. ^ Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, pp. 261 ff .
2. ^ Roger A. Horn, Charles R. Johnson: Matrix Analysis . Cambridge University Press, 2012, pp. 300 ff .

Web links

• Eric W. Weisstein : Conjugate Matrix . In: MathWorld (English).