Normal matrix

In linear algebra, a normal matrix is a matrix with the property ${\displaystyle A\in \mathbb {C} ^{n\times n}}$

${\displaystyle A^{*}\cdot A=A\cdot A^{*}}$,

thus a matrix that commutes with its adjoint matrix . Similarly, a real matrix is normal if ${\displaystyle B\in \mathbb {R} ^{n\times n}}$

${\displaystyle B^{T}\cdot B=B\cdot B^{T}}$

applies.

The spectral theorem says that a matrix is normal if and only if there is a unitary matrix such that , where is a diagonal matrix . Normal matrices therefore have the property that they can be unitarily diagonalized . There is therefore an orthonormal basis made up of eigenvectors of . The main diagonal elements of are exactly the eigenvalues of . In particular, every real symmetric matrix and every complex Hermitian matrix are normal. In addition, every unitary matrix is ​​normal. ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle A=UDU^{\rm {*}}}$ ${\displaystyle D}$ ${\displaystyle A}$ ${\displaystyle D}$ ${\displaystyle A}$

Examples

The eigenvalues ​​can be complex even if the matrix is real, and so are generally complex, as the example shows: ${\displaystyle A}$ ${\displaystyle U}$ ${\displaystyle D}$

${\displaystyle A={\begin{pmatrix}0&1\\-1&0\\\end{pmatrix}}\implies U={\tfrac {1}{\sqrt {2}}}{\begin{pmatrix}1&1\\i&-i\\\end{pmatrix}},\quad D={\begin{pmatrix}i&0\\0&-i\\\end{pmatrix}}}$.

Only in the special case of a real symmetric matrix are the matrix and the eigenvalues ​​(i.e. ) always real. ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle U}$ ${\displaystyle D}$

It should be noted that there are matrices that are diagonalizable but not normal. In this case, no unitary diagonalizability is present, that is, it applies only where not is unitary, that is . An example of a non-normal but diagonalizable matrix is ${\displaystyle A=TDT^{-1}}$ ${\displaystyle T}$ ${\displaystyle T^{-1}\neq T^{*}}$

${\displaystyle A={\begin{pmatrix}0&1\\4&0\\\end{pmatrix}}}$.

Normality and deviations from normality

The decomposition of the matrix into is also called the Schur decomposition or Schur's normal form. Basically: ${\displaystyle A}$ ${\displaystyle UDU^{*}}$

${\displaystyle UDU^{*}=\operatorname {diag} \{\lambda _{1},\dotsc ,\lambda _{n}\}+N}$,

where is a strict upper triangular matrix (there are only zeros on the diagonal) and the eigenvalues ​​of are. The following applies to normal matrices: ${\displaystyle N}$ ${\displaystyle \lambda _{1}>\lambda _{2}>\dotsb >\lambda _{n}}$ ${\displaystyle A}$

${\displaystyle \|N\|_{F}=0}$.

Is not normal, it is called the deviation from normal. The norm denotes the Frobenius norm . ${\displaystyle A}$ ${\displaystyle \|N\|_{F}=:\delta (A)}$ ${\displaystyle \|\cdot \|_{F}}$

Normal matrices and normal operators

→ Main article : Normal operator

A normal operator is a generalization of the normal matrix in two ways:

1. A normal matrix describes a normal operator with respect to a suitable basis (namely with respect to an orthonormal basis ), while the term "normal operator" is defined independently of the basis,
2. Normal matrices describe normal operators on finite-dimensional scalar product spaces , while normal operators are also (and even mostly) used on infinite-dimensional spaces.

The basic dependence of the term "normal" for a matrix comes into play through the definition of "adjoint": The matrix to be adjoint is defined by the following property: ${\displaystyle A}$ ${\displaystyle A^{*}}$

${\displaystyle \langle Av,w\rangle =\langle v,A^{*}\,w\rangle }$for everyone . ${\displaystyle v,w\in \mathbb {K} ^{n}}$

This definition can also be read independently of the basis, but only if the vectors in this definition are coordinate vectors with respect to an orthonormal basis, the scalar product can be written as a matrix product (see also matrix (mathematics) # vector spaces of matrices ), so that follows for any matrices : ${\displaystyle A}$

${\displaystyle \langle A\cdot v,w\rangle ={\overline {(A\cdot v)}}^{T}\cdot w=({\overline {v}}^{T}\cdot {\overline {A}}^{T})\cdot w={\overline {v}}^{T}\cdot ({\overline {A}}^{T}\cdot w)=\langle v,{\overline {A}}^{T}\cdot w\rangle .}$

Only then can the matrix to be adjoint always be calculated by conjugation and transposition. ${\displaystyle A}$

literature

• Gerd Fischer : Linear Algebra. (An introduction for first-year students). 13th revised edition. Vieweg, Braunschweig et al. 2002, ISBN 3-528-97217-3 .