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Hermitian matrix

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From Wikipedia, the free encyclopedia

Matrix equal to its conjugate-transpose

For matrices with symmetry over thereal number field, seeSymmetric matrix.

Inmathematics, aHermitian matrix (orself-adjoint matrix) is asquare matrix that is equal to its ownconjugate transpose—that is, the element in thei-th row andj-th column is equal to thecomplex conjugate of the element in thej-th row andi-th column, for all indicesi andj. Inindex form, or in matrix form: $A{\text{ is Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}$

Hermitian matrices can be understood as the complex extension of realsymmetric matrices.

If theconjugate transpose of a matrix $A {\displaystyle A}$ is denoted by $A^{\mathsf {H}},$ then the Hermitian property can be written concisely as

$A{\text{ is Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}$

Hermitian matrices are named afterCharles Hermite,^[1] who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having realeigenvalues. Other, equivalent notations in common use are $A^{\mathsf {H}}=A^{\dagger }=A^{\ast },$ although inquantum mechanics, $A^{\ast }$ typically means thecomplex conjugate only, and not theconjugate transpose.

Alternative characterizations

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Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:

Equality with the adjoint

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A square matrix $A {\displaystyle A}$ is Hermitian if and only if it is equal to itsconjugate transpose, that is, it satisfies $\langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,$ for any pair of vectors $\mathbf {v} ,\mathbf {w} ,$ where $\langle \cdot ,\cdot \rangle$ denotesthe inner product operation.

This is also the way that the more general concept ofself-adjoint operator is defined.

Real-valuedness of quadratic forms

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An $n\times {}n$ matrix $A {\displaystyle A}$ is Hermitian if and only if $\langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad {\text{for all }}\mathbf {v} \in \mathbb {C} ^{n}.$

Spectral properties

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A square matrix $A {\displaystyle A}$ is Hermitian if and only if it is unitarilydiagonalizable with realeigenvalues.

Applications

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Hermitian matrices are fundamental toquantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue $a {\displaystyle a}$ of an operator ${\hat {A}}$ on some quantum state $|\psi \rangle$ is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues.

Insignal processing, Hermitian matrices are utilized in tasks likeFourier analysis and signal representation.^[2] The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information.

Hermitian matrices are extensively studied inlinear algebra andnumerical analysis. They have well-defined spectral properties, and many numerical algorithms, such as theLanczos algorithm, exploit these properties for efficient computations. Hermitian matrices also appear in techniques likesingular value decomposition (SVD) andeigenvalue decomposition.

Instatistics andmachine learning, Hermitian matrices are used incovariance matrices, where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.^[3]

Hermitian matrices are applied in the design and analysis ofcommunications system, especially in the field ofmultiple-input multiple-output (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties.

Ingraph theory, Hermitian matrices are used to study thespectra of graphs. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.^[4] The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs.^[5]

Examples and solutions

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In this section, the conjugate transpose of matrix $A {\displaystyle A}$ is denoted as $A^{\mathsf {H}},$ the transpose of matrix $A {\displaystyle A}$ is denoted as $A^{\mathsf {T}}$ and conjugate of matrix $A {\displaystyle A}$ is denoted as ${\overline {A}}.$

See the following example:

${\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}$

The diagonal elements must bereal, as they must be their own complex conjugate.

Well-known families of Hermitian matrices include thePauli matrices, theGell-Mann matrices and their generalizations. Intheoretical physics such Hermitian matrices are often multiplied byimaginary coefficients,^[6]^[7] which results inskew-Hermitian matrices.

Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix $A {\displaystyle A}$ equals theproduct of a matrix with its conjugate transpose, that is, $A=BB^{\mathsf {H}},$ then $A {\displaystyle A}$ is a Hermitianpositive semi-definite matrix. Furthermore, if $B {\displaystyle B}$ is row full-rank, then $A {\displaystyle A}$ is positive definite.

Properties

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Main diagonal values are real

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The entries on themain diagonal (top left to bottom right) of any Hermitian matrix arereal.

Proof

By definition of the Hermitian matrix $H_{ij}={\overline {H}}_{ji}$ so fori =j the above follows, as a number can equal its complex conjugate only if the imaginary parts are zero.

Only themain diagonal entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in theiroff-diagonal elements, as long as diagonally-opposite entries are complex conjugates.

Symmetric

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A matrix that has only real entries issymmetric if and only if it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.

Proof

$H_{ij}={\overline {H}}_{ji}$ by definition. Thus $H_{ij}=H_{ji}$ (matrix symmetry) if and only if $H_{ij}={\overline {H}}_{ij}$ ( $H_{ij}$ is real).

So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit $i, {\displaystyle i,}$ then it becomes Hermitian.

Normal

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Every Hermitian matrix is anormal matrix. That is to say, $AA^{\mathsf {H}}=A^{\mathsf {H}}A.$

Proof

$A=A^{\mathsf {H}},$ so $AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.$

Diagonalizable

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The finite-dimensionalspectral theorem says that any Hermitian matrix can bediagonalized by aunitary matrix, and that the resulting diagonal matrix has only real entries. This implies that alleigenvalues of a Hermitian matrixA with dimensionn are real, and thatA hasn linearly independenteigenvectors. Moreover, a Hermitian matrix hasorthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find anorthogonal basis ofCⁿ consisting ofn eigenvectors ofA.

Sum of Hermitian matrices

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The sum of any two Hermitian matrices is Hermitian.

Proof

$(A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},$ as claimed.

Inverse is Hermitian

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Theinverse of an invertible Hermitian matrix is Hermitian as well.

Proof

If $A^{-1}A=I,$ then $I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {H}},$ so $A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}$ as claimed.

Associative product of Hermitian matrices

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Theproduct of two Hermitian matricesA andB is Hermitian if and only ifAB =BA.

Proof

$(AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}\ {\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.$ Thus $(AB)^{\mathsf {H}}=AB$ if and only if $AB=BA.$

ThusAⁿ is Hermitian ifA is Hermitian andn is an integer.

ABA Hermitian

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IfA andB are Hermitian, thenABA is also Hermitian.

Proof

$(ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA$

v^HAv is real for complexv

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For an arbitrary complex valued vectorv the product $\mathbf {v} ^{\mathsf {H}}A\mathbf {v}$ is real because of $\mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.$ This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system, e.g. totalspin, which have to be real.

Complex Hermitian forms vector space overℝ

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The Hermitian complexn-by-n matrices do not form avector space over thecomplex numbers,ℂ, since the identity matrixI_n is Hermitian, buti I_n is not. However the complex Hermitian matricesdo form a vector space over thereal numbersℝ. In the2n²-dimensional vector space of complexn × n matrices overℝ, the complex Hermitian matrices form a subspace of dimensionn². IfE_jk denotes then-by-n matrix with a1 in thej,k position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows: $E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})$

together with the set of matrices of the form ${\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)$

and the matrices ${\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)$

where $i {\displaystyle i}$ denotes theimaginary unit, $i={\sqrt {-1}}~.$

An example is that the fourPauli matrices form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices overℝ.

Eigendecomposition

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Ifn orthonormal eigenvectors $\mathbf {u} _{1},\dots ,\mathbf {u} _{n}$ of a Hermitian matrix are chosen and written as the columns of the matrixU, then oneeigendecomposition ofA is $A=U\Lambda U^{\mathsf {H}}$ where $UU^{\mathsf {H}}=I=U^{\mathsf {H}}U$ and therefore $A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {u} _{j}^{\mathsf {H}},$ where $\lambda _{j}$ are the eigenvalues on the diagonal of the diagonal matrix $\Lambda .$

Singular values

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The singular values of $A {\displaystyle A}$ are the absolute values of its eigenvalues:

Since $A {\displaystyle A}$ has an eigendecomposition $A=U\Lambda U^{H}$ , where $U {\displaystyle U}$ is aunitary matrix (its columns are orthonormal vectors;see above), asingular value decomposition of $A {\displaystyle A}$ is $A=U|\Lambda |{\text{sgn}}(\Lambda )U^{H}$ , where $|\Lambda |$ and ${\text{sgn}}(\Lambda )$ are diagonal matrices containing the absolute values $|\lambda |$ and signs ${\text{sgn}}(\lambda )$ of $A {\displaystyle A}$ 's eigenvalues, respectively. $\operatorname {sgn}(\Lambda )U^{H}$ is unitary, since the columns of $U^{H}$ are only getting multiplied by $\pm 1$ . $|\Lambda |$ contains the singular values of $A {\displaystyle A}$ , namely, the absolute values of its eigenvalues.^[8]

Real determinant

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The determinant of a Hermitian matrix is real:

Proof

$\det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}$ Therefore if $A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}.$

(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

Decomposition into Hermitian and skew-Hermitian matrices

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Additional facts related to Hermitian matrices include:

The sum of a square matrix and its conjugate transpose $\left(A+A^{\mathsf {H}}\right)$ is Hermitian.
The difference of a square matrix and its conjugate transpose $\left(A-A^{\mathsf {H}}\right)$ isskew-Hermitian (also called antihermitian). This implies that thecommutator of two Hermitian matrices is skew-Hermitian.
An arbitrary square matrixC can be written as the sum of a Hermitian matrixA and a skew-Hermitian matrixB. This is known as the Toeplitz decomposition ofC.^[9]^: 227 $C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)$

Rayleigh quotient

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Main article:Rayleigh quotient

In mathematics, for a given complex Hermitian matrixM and nonzero vectorx, the Rayleigh quotient^[10] $R(M,\mathbf {x} ),$ is defined as:^[9]^{: p. 234}^[11] $R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.$

For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose $\mathbf {x} ^{\mathsf {H}}$ to the usual transpose $\mathbf {x} ^{\mathsf {T}}.$ $R(M,c\mathbf {x} )=R(M,\mathbf {x} )$ for any non-zero real scalar $c . {\displaystyle c.}$ Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.

It can be shown^[9] that, for a given matrix, the Rayleigh quotient reaches its minimum value $\lambda _{\min }$ (the smallest eigenvalue of M) when $\mathbf {x}$ is $\mathbf {v} _{\min }$ (the corresponding eigenvector). Similarly, $R(M,\mathbf {x} )\leq \lambda _{\max }$ and $R(M,\mathbf {v} _{\max })=\lambda _{\max }.$

The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.

The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, $\lambda _{\max }$ is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that toM associates the Rayleigh quotientR(M,x) for a fixedx andM varying through the algebra would be referred to as "vector state" of the algebra.

References

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^Archibald, Tom (2010-12-31), Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.),"VI.47 Charles Hermite",The Princeton Companion to Mathematics, Princeton University Press, p. 773,doi:10.1515/9781400830398.773a,ISBN 978-1-4008-3039-8, retrieved2023-11-15{{citation}}: CS1 maint: work parameter with ISBN (link)
^Ribeiro, Alejandro."Signal and Information Processing"(PDF).
^"MULTIVARIATE NORMAL DISTRIBUTIONS"(PDF).
^Lau, Ivan."Hermitian Spectral Theory of Mixed Graphs"(PDF).
^Liu, Jianxi; Li, Xueliang (February 2015)."Hermitian-adjacency matrices and Hermitian energies of mixed graphs".Linear Algebra and Its Applications.466:182–207.doi:10.1016/j.laa.2014.10.028.
^Frankel, Theodore (2004).The Geometry of Physics: an introduction.Cambridge University Press. p. 652.ISBN 0-521-53927-7.
^Physics 125 Course Notes Archived 2022-03-07 at theWayback Machine atCalifornia Institute of Technology
^Trefethan, Lloyd N.; Bau, III, David (1997).Numerical linear algebra. Philadelphia, PA, USA:SIAM. p. 34.ISBN 0-89871-361-7.OCLC 1348374386.
^^a ^b ^cHorn, Roger A.; Johnson, Charles R. (2013).Matrix Analysis, second edition. Cambridge University Press.ISBN 9780521839402.
^Also known as theRayleigh–Ritz ratio; named afterWalther Ritz andLord Rayleigh.
^Parlett, Beresford N. (1998).The symmetric eigenvalue problem. Classics in applied mathematics. Philadelphia, Pa: Society for Industrial and Applied Mathematics.ISBN 978-1-61197-116-3.

External links

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"Hermitian matrix",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Visualizing Hermitian Matrix as An Ellipse with Dr. Geo Archived 2017-08-29 at theWayback Machine, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
"Hermitian Matrices".MathPages.com.

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