View source for Hermitian matrix

{{Short description|Matrix equal to its conjugate-transpose
}}

In [[Mathematics|mathematics]], a '''Hermitian matrix''' (or '''self-adjoint matrix''') is a [[Complex number|complex]] [[Square matrix|square matrix]] that is equal to its own [[Conjugate transpose|conjugate transpose]]—that is, the element in the {{mvar|i}}-th row and {{mvar|j}}-th column is equal to the [[Complex conjugate|complex conjugate]] of the element in the {{mvar|j}}-th row and {{mvar|i}}-th column, for all indices {{mvar|i}} and {{mvar|j}}:

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|equation = <math>A \text{ Hermitian} \quad \iff \quad a_{ij} = \overline{{a}_{ji}}</math>
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or in matrix form:
<math display=block>A \text{ Hermitian} \quad \iff \quad A = \overline {A^\mathsf{T}}.</math>

Hermitian matrices can be understood as the complex extension of real [[Symmetric matrix|symmetric matrices]].

If the [[Conjugate transpose|conjugate transpose]] of a matrix <math>A</math> is denoted by <math>A^\mathsf{H},</math> then the Hermitian property can be written concisely as

{{Equation box 1
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|equation = <math>A \text{ Hermitian} \quad \iff \quad A = A^\mathsf{H}</math>
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Hermitian matrices are named after [[Biography:Charles Hermite|Charles Hermite]],<ref>{{Citation |last=Archibald |first=Tom |title=VI.47 Charles Hermite |date=2010-12-31 |url=https://www.degruyter.com/document/doi/10.1515/9781400830398.773a/html |work=The Princeton Companion to Mathematics |pages=773–773 |editor-last=Gowers |editor-first=Timothy |access-date=2023-11-15 |publisher=Princeton University Press |doi=10.1515/9781400830398.773a |isbn=978-1-4008-3039-8 |editor2-last=Barrow-Green |editor2-first=June |editor3-last=Leader |editor3-first=Imre}}</ref> who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real [[Eigenvalues and eigenvectors|eigenvalues]]. Other, equivalent notations in common use are <math>A^\mathsf{H} = A^\dagger = A^\ast,</math> although in [[Physics:Quantum mechanics|quantum mechanics]], <math>A^\ast</math> typically means the [[Complex conjugate|complex conjugate]] only, and not the [[Conjugate transpose|conjugate transpose]].

==Alternative characterizations==

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

===Equality with the adjoint===

A square matrix <math>A</math> is Hermitian if and only if it is equal to its [[Hermitian adjoint|conjugate transpose]], that is, it satisfies
<math display="block">\langle \mathbf w, A \mathbf v\rangle = \langle A \mathbf w, \mathbf v\rangle,</math>
for any pair of vectors <math>\mathbf v, \mathbf w,</math> where <math>\langle \cdot, \cdot\rangle</math> denotes [[Dot product|the inner product]] operation.

This is also the way that the more general concept of [[Self-adjoint operator|self-adjoint operator]] is defined.

===Real-valuedness of quadratic forms===

An <math>n\times{}n</math> matrix <math>A</math> is Hermitian if and only if
<math display="block">\langle \mathbf{v}, A \mathbf{v}\rangle\in\R, \quad \mathbf{v}\in \mathbb{C}^{n}.</math>

===Spectral properties===

A square matrix <math>A</math> is Hermitian if and only if it is unitarily [[Diagonalizable matrix|diagonalizable]] with real [[Eigenvalues and eigenvectors|eigenvalues]].'''

==Applications==

Hermitian matrices are fundamental to [[Physics:Quantum mechanics|quantum mechanics]] because they describe operators with necessarily real eigenvalues. An eigenvalue <math>a</math> of an operator <math>\hat{A}</math> on some quantum state <math>|\psi\rangle</math> is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues.

In [[Signal processing|signal processing]], Hermitian matrices are utilized in tasks like [[Fourier analysis]] and signal representation.<ref>{{Cite web |last=Ribeiro |first=Alejandro |title=Signal and Information Processing |url=https://www.seas.upenn.edu/~ese2240/wiki/Lecture%20Notes/sip_PCA.pdf}}</ref> The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information.

Hermitian matrices are extensively studied in [[Linear algebra|linear algebra]] and [[Numerical analysis|numerical analysis]]. They have well-defined spectral properties, and many numerical algorithms, such as the [[Lanczos algorithm]], exploit these properties for efficient computations. Hermitian matrices also appear in techniques like [[Singular value decomposition|singular value decomposition]] (SVD) and eigenvalue decomposition.

In [[Statistics|statistics]] and [[Machine learning|machine learning]], Hermitian matrices are used in [[Covariance matrix|covariance matrices]], where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.<ref>{{Cite web |title=MULTIVARIATE NORMAL DISTRIBUTIONS |url=https://dspace.mit.edu/bitstream/handle/1721.1/121170/6-436j-fall-2008/contents/lecture-notes/MIT6_436JF08_lec15.pdf}}</ref>

Hermitian matrices are applied in the design and analysis of [[Social:Communications system|communications system]], especially in the field of multiple-input multiple-output (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties.

In [[Graph theory|graph theory]], Hermitian matrices are used to study the [[Spectral graph theory|spectra of graphs]]. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.<ref>{{Cite web |last=Lau |first=Ivan |title=Hermitian Spectral Theory of Mixed Graphs |url=https://www.sfu.ca/~iplau/Edinburgh_CS_Project.pdf}}</ref> 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.<ref>{{Cite journal |last=Liu |first=Jianxi |last2=Li |first2=Xueliang |date=February 2015 |title=Hermitian-adjacency matrices and Hermitian energies of mixed graphs |url=https://linkinghub.elsevier.com/retrieve/pii/S0024379514006971 |journal=Linear Algebra and its Applications |language=en |volume=466 |pages=182–207 |doi=10.1016/j.laa.2014.10.028|doi-access=free }}</ref>

==Examples and solutions==

In this section, the conjugate transpose of matrix <math> A </math> is denoted as <math> A^\mathsf{H} ,</math> the transpose of matrix <math> A </math> is denoted as <math> A^\mathsf{T} </math> and conjugate of matrix <math> A </math> is denoted as <math> \overline{A} .</math>

See the following example:

<math display=block>\begin{bmatrix}
  0     & a - ib & c-id \\
  a+ib  & 1     & m-in \\
  c+id     &   m+in & 2
\end{bmatrix}</math>

The diagonal elements must be [[Real number|real]], as they must be their own complex conjugate.

Well-known families of Hermitian matrices include the [[Pauli matrices]], the [[Physics:Gell-Mann matrices|Gell-Mann matrices]] and their generalizations. In [[Physics:Theoretical physics|theoretical physics]] such Hermitian matrices are often multiplied by [[Imaginary number|imaginary]] coefficients,<ref>
{{cite book |title=The Geometry of Physics: an introduction |last=Frankel |first=Theodore |year=2004 |publisher={{wipe|Cambridge University Press}} |isbn=0-521-53927-7 |page=652 |url=https://books.google.com/books?id=DUnjs6nEn8wC&q=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&pg=PA652 }}
</ref><ref>[http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf Physics 125 Course Notes] at [[Organization:California Institute of Technology|California Institute of Technology]]</ref> which results in [[Skew-Hermitian matrix|skew-Hermitian matrices]].

Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix <math> A </math> equals the [[Matrix multiplication|product of a matrix]] with its conjugate transpose, that is, <math> A = BB^\mathsf{H} ,</math> then <math> A </math> is a Hermitian positive semi-definite matrix. Furthermore, if <math> B </math> is row full-rank, then <math> A </math> is positive definite.

==Properties==

===Main diagonal values are real===

The entries on the [[Main diagonal|main diagonal]] (top left to bottom right) of any Hermitian matrix are [[Real number|real]].

{{math proof|1= By definition of the Hermitian matrix
<math display=block>H_{ij} = \overline{H}_{ji} </math>
so for {{math|1=''i'' = ''j''}} the above follows.
}}

Only the [[Main diagonal|main diagonal]] entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their off-diagonal elements, as long as diagonally-opposite entries are complex conjugates.

===Symmetric===

A matrix that has only real entries is [[Symmetric matrix|symmetric]] [[If and only if|if and only if]] it is Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix.

{{math proof|1= <math>H_{ij} = \overline{H}_{ji}</math> by definition. Thus  <math>H_{ij} = H_{ji}</math> (matrix symmetry) if and only if <math>H_{ij} = \overline{H}_{ij}</math> (<math>H_{ij}</math> is real).
}}

So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit <math>i,</math> then it becomes Hermitian.

===Normal===

Every Hermitian matrix is a [[Normal matrix|normal matrix]]. That is to say, <math>AA^\mathsf{H} = A^\mathsf{H}A.</math>

{{math proof|1=<math>A = A^\mathsf{H},</math> so <math>AA^\mathsf{H} = AA = A^\mathsf{H}A.</math>}}

===Diagonalizable===

The finite-dimensional [[Spectral theorem|spectral theorem]] says that any Hermitian matrix can be [[Diagonalizable matrix|diagonalized]] by a [[Unitary matrix|unitary matrix]], and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix {{mvar|A}} with dimension {{mvar|n}} are real, and that {{mvar|A}} has {{mvar|n}} linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an [[Orthogonal basis|orthogonal basis]] of {{math|'''C'''<sup>''n''</sup>}} consisting of {{mvar|n}} eigenvectors of {{mvar|A}}.

===Sum of Hermitian matrices===

The sum of any two Hermitian matrices is Hermitian.

{{math proof|1= <math display="block">(A + B)_{ij} = A_{ij} + B_{ij} = \overline{A}_{ji} + \overline{B}_{ji} = \overline{(A + B)}_{ji},</math> as claimed.}}

===Inverse is Hermitian===

The inverse of an invertible Hermitian matrix is Hermitian as well.

{{math proof|1= If <math>A^{-1}A = I,</math> then <math>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},</math> so <math>A^{-1}=\left(A^{-1}\right)^\mathsf{H}</math> as claimed.}}

===Associative product of Hermitian matrices===

The [[Matrix multiplication|product]] of two Hermitian matrices {{mvar|A}} and {{mvar|B}} is Hermitian if and only if {{math|1=''AB'' = ''BA''}}.

{{math proof|1= <math display="block">(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.</math> Thus <math>(AB)^\mathsf{H} = AB</math> [[If and only if|if and only if]] <math>AB = BA.</math>

Thus {{math|''A''<sup>''n''</sup>}} is Hermitian if {{mvar|A}} is Hermitian and {{mvar|n}} is an integer.
}}

===''ABA'' Hermitian===

If ''A'' and ''B'' are Hermitian, then ''ABA'' is also Hermitian.
{{math proof|1= <math display="block">(ABA)^\mathsf{H} = (A(BA))^\mathsf{H} = (BA)^\mathsf{H}A^\mathsf{H} = A^\mathsf{H}B^\mathsf{H}A^\mathsf{H} = ABA </math>}}

==={{math|v<sup>H</sup>''A''v}} is real for complex {{math|v}}===

For an arbitrary complex valued vector {{Math|'''v'''}} the product <math> \mathbf{v}^\mathsf{H} A \mathbf{v} </math> is real because of <math> \mathbf{v}^\mathsf{H} A \mathbf{v} = \left(\mathbf{v}^\mathsf{H} A \mathbf{v}\right)^\mathsf{H} .</math> This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system, e.g. total [[Physics:Spin|spin]], which have to be real.

===Complex Hermitian forms vector space over {{math|R}}===

The Hermitian complex {{mvar|n}}-by-{{mvar|n}} matrices do not form a [[Vector space|vector space]] over the [[Complex number|complex number]]s, {{math|'''C'''}}, since the identity matrix {{math|''I''<sub>''n''</sub>}} is Hermitian, but {{math|''i'' ''I''<sub>''n''</sub>}} is not. However the complex Hermitian matrices ''do'' form a vector space over the real numbers {{math|'''R'''}}. In the {{math|2''n''<sup>2</sup>}}-dimensional vector space of complex {{math|''n'' × ''n''}} matrices over {{math|'''R'''}}, the complex Hermitian matrices form a subspace of dimension {{math|''n''<sup>2</sup>}}. If {{math|''E''<sub>''jk''</sub>}} denotes the {{mvar|n}}-by-{{mvar|n}} matrix with a {{math|1}} in the {{math|''j'',''k''}} position and zeros elsewhere, a basis (orthonormal with respect to the Frobenius inner product) can be described as follows:
<math display=block>E_{jj} \text{ for } 1 \leq j \leq n \quad (n \text{ matrices}) </math>

together with the set of matrices of the form
<math display=block>\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) </math>

and the matrices
<math display=block>\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) </math>

where <math>i</math> denotes the [[Imaginary unit|imaginary unit]], <math>i = \sqrt{-1}~.</math>

An example is that the four [[Pauli matrices]] form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over {{math|'''R'''}}.

===Eigendecomposition===

If {{mvar|n}} orthonormal eigenvectors <math>\mathbf{u}_1, \dots, \mathbf{u}_n</math> of a Hermitian matrix are chosen and written as the columns of the matrix {{mvar|U}}, then one [[Eigendecomposition of a matrix|eigendecomposition]] of {{mvar|A}} is <math> A = U \Lambda U^\mathsf{H}</math> where <math>U U^\mathsf{H} = I = U^\mathsf{H} U</math> and therefore 
<math display=block>A = \sum_j \lambda_j \mathbf{u}_j \mathbf{u}_j ^\mathsf{H},</math>
where <math>\lambda_j</math> are the eigenvalues on the diagonal of the diagonal matrix <math>\Lambda.</math>

=== Singular values ===
The singular values of <math>A</math> are the absolute values of its eigenvalues:

Since <math>A</math> has an eigendecomposition <math>A=U\Lambda U^H</math>, where <math>U</math> is a [[Unitary matrix|unitary matrix]] (its columns are orthonormal vectors; [[Hermitian matrix#Eigendecomposition|see above]]), a [[Singular value decomposition|singular value decomposition]] of <math>A</math> is <math>A=U|\Lambda|\text{sgn}(\Lambda)U^H</math>, where <math>|\Lambda|</math> and <math>\text{sgn}(\Lambda)</math> are diagonal matrices containing the absolute values <math>|\lambda|</math> and signs <math>\text{sgn}(\lambda)</math> of <math>A</math>'s eigenvalues, respectively. <math>\sgn(\Lambda)U^H</math> is unitary, since the columns of <math>U^H</math> are only getting multiplied by <math>\pm 1</math>. <math>|\Lambda|</math> contains the singular values of <math>A</math>, namely, the absolute values of its eigenvalues.<ref>{{Cite book |last=Trefethan |first=Lloyd N. |url=http://worldcat.org/oclc/1348374386 |title=Numerical linear algebra |last2=Bau, III |first2=David |publisher=SIAM |year=1997 |isbn=0-89871-361-7 |location=Philadelphia, PA, USA |pages=34 |oclc=}}</ref>

===Real determinant===

The determinant of a Hermitian matrix is real:

{{math proof|1= <math> \det(A) = \det\left(A^\mathsf{T}\right)\quad \Rightarrow \quad \det\left(A^\mathsf{H}\right) = \overline{\det(A)} </math>
Therefore if <math>A = A^\mathsf{H}\quad \Rightarrow \quad \det(A) = \overline{\det(A)} .</math>
}}
(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==

{{anchor|facts}}Additional facts related to Hermitian matrices include:
* The sum of a square matrix and its conjugate transpose <math>\left(A + A^\mathsf{H}\right)</math> is Hermitian.
* The difference of a square matrix and its conjugate transpose <math>\left(A - A^\mathsf{H}\right)</math> is [[Skew-Hermitian matrix|skew-Hermitian]] (also called antihermitian). This implies that the [[Commutator|commutator]] of two Hermitian matrices is skew-Hermitian.
* An arbitrary square matrix {{mvar|C}} can be written as the sum of a Hermitian matrix {{mvar|A}} and a skew-Hermitian matrix {{mvar|B}}. This is known as the Toeplitz decomposition of {{mvar|C}}.<ref name="HornJohnson">{{cite book |title=Matrix Analysis, second edition |first1=Roger A. |last1=Horn |first2=Charles R. |last2=Johnson |isbn=9780521839402 |publisher=Cambridge University Press|year=2013}}</ref>{{rp|227}} <math display="block">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)</math>

==Rayleigh quotient==
{{Main|Rayleigh quotient}}

In mathematics, for a given complex Hermitian matrix {{mvar|M}} and nonzero vector {{math|'''x'''}}, the Rayleigh quotient<ref>Also known as the '''Rayleigh–Ritz ratio'''; named after Walther Ritz and Lord Rayleigh.</ref> <math>R(M, \mathbf{x}),</math> is defined as:<ref name="HornJohnson"/>{{rp|p. 234}}<ref>Parlet B. N. ''The symmetric eigenvalue problem'', SIAM, Classics in Applied Mathematics,1998</ref>
<math display=block>R(M, \mathbf{x}) := \frac{\mathbf{x}^\mathsf{H} M \mathbf{x}}{\mathbf{x}^\mathsf{H} \mathbf{x}}.</math>

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

It can be shown<ref name="HornJohnson" /> that, for a given matrix, the Rayleigh quotient reaches its minimum value <math>\lambda_\min</math> (the smallest eigenvalue of M) when <math>\mathbf x</math> is <math>\mathbf v_\min</math> (the corresponding eigenvector). Similarly, <math>R(M, \mathbf x) \leq \lambda_\max</math> and <math>R(M, \mathbf v_\max) = \lambda_\max .</math>

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, <math>\lambda_\max</math> is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to {{math|''M''}} associates the Rayleigh quotient {{math|''R''(''M'', ''x'')}} for a fixed {{math|'''x'''}} and {{math|''M''}} varying through the algebra would be referred to as "vector state" of the algebra.

==See also==

* {{annotated link|Haynsworth inertia additivity formula}}
* {{annotated link|Normal matrix}}
* {{annotated link|Schur–Horn theorem}}
* {{annotated link|Self-adjoint operator}}
* {{annotated link|Skew-Hermitian matrix}} (anti-Hermitian matrix)
* {{annotated link|Unitary matrix}}
* {{annotated link|Vector space}}

==References==

{{reflist}}

==External links==
* {{springer|title=Hermitian matrix|id=p/h047070}}
* [https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php Visualizing Hermitian Matrix as An Ellipse with Dr. Geo], by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
*{{MathPages|id=home/kmath306/kmath306|title=Hermitian Matrices}}

{{Matrix classes}}

[[Category:Matrices]]

{{Sourceattribution|Hermitian matrix}}