Hermitian matrix

For symmetric real or complex matrices, see Symmetric matrix.

In mathematics, more precisely in linear algebra, a Hermitian matrix (or self-adjoint matrix) is a square matrix that is equal to its own conjugate transpose—that is, its element in the i-th row and j-th column is the complex conjugate of its element in the j-th row and i-th column, for all indices i and j. With matrix notations:

$$A{\text{ is Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}\quad \iff \quad A_{ij}={\overline {A_{ji}}},$$

where $A^{\mathsf {T}}$ is the transpose of $A,~$ ${\overline {B}}$ is the conjugate of $B,$ and $A_{ij}$ is the element in the i-th row and j-th column of $A.$

Hermitian matrices can be understood as the complex generalization of symmetric real matrices.

The Hermitian property of a matrix $A$ can be written concisely as

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

where $A^{\mathsf {H}}$ denotes the conjugate transpose of $A.$

Equivalent notations in common use are $A^{\mathsf {H}}=A^{\dagger }=A^{*},$ although in quantum mechanics, $A^{*}$ typically means the complex conjugate only, and not the conjugate transpose.

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share with symmetric real matrices the property of always having real eigenvalues.

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 $A$ is Hermitian if and only if it is equal to its conjugate transpose, that is, for any pair of vectors $\mathbf {v} ,\mathbf {w} ,$ it satisfies

$$\langle \mathbf {v} ,A\mathbf {w} \rangle =\langle A\mathbf {v} ,\mathbf {w} \rangle ,$$

where $\langle \cdot ,\cdot \rangle$ denotes the inner product operation.

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

Real-valuedness of quadratic forms

An $n\times n$ matrix $A$ 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

A square matrix is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues.

Applications

Hermitian matrices are fundamental to quantum mechanics because they describe operators with necessarily real eigenvalues. An eigenvalue $a$ 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.

In signal processing, Hermitian matrices are utilized in tasks like Fourier analysis and signal representation. 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 and 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 (SVD) and eigenvalue decomposition.

In statistics and machine learning, Hermitian matrices are used as covariance matrices, which represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.

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

In graph theory, Hermitian matrices are used to study the 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. 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.

Examples

In the following example, $\mathbf {i}$ denotes the imaginary unit $\left(\mathbf {i} ={\sqrt {-1}}\right)\!:$

$${\begin{bmatrix}0&a-\mathbf {i} b&c-\mathbf {i} d\\a+\mathbf {i} b&1&m-\mathbf {i} n\\c+\mathbf {i} d&m+\mathbf {i} n&2\end{bmatrix}}.$$

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

Well-known families of Hermitian matrices include the Pauli matrices, the Gell-Mann matrices, and their generalizations. In theoretical physics, such Hermitian matrices are often multiplied by imaginary coefficients, which results in skew-Hermitian matrices.

Here is another useful case of a Hermitian matrix. If a square matrix $A$ equals the product of a matrix $B$ with its conjugate transpose, that is, $A=BB^{\mathsf {H}},$ then $A$ is Hermitian positive semi-definite. Furthermore, if $B$ is non-singular (i.e., row full-rank), then $A$ is positive definite.

Properties

Main diagonal values are real

The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are real.

Proof

$H_{ij}={\overline {H_{ji}}}$ by definition of a Hermitian matrix; so $H_{ii}={\overline {H_{ii}}},$ and the above follows, as a number can equal its complex conjugate only if their imaginary parts are zero.

Only the main diagonal entries are necessarily real; a Hermitian matrix can have arbitrary complex-valued entries as its off-diagonal elements, as long as its diagonally-opposite entries are complex conjugates.

Symmetric if and only if real

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

Proof

$H_{ij}={\overline {H_{ji}}}$ by definition of a Hermitian matrix. Thus,
$H_{ij}=H_{ji}$ (i.e., $H$ is symmetric) if and only if ${\overline {H_{ji}}}=H_{ji}$ (i.e., $H_{ji}$ is real).

Related special case

If an anti-symmetric real matrix is multiplied by an imaginary number, then the product is also anti-symmetric but has only imaginary entries. An anti-symmetric imaginary matrix is another special case of a Hermitian matrix.

Normal

Every Hermitian matrix $A$ is a normal 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

The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix A with order n are real, and that A has n linearly independent eigenvectors. Moreover, a Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues (i.e., zero eigenvalues), it is always possible to find an orthonormal basis of $\mathbb {C} ^{n}$ consisting of n eigenvectors of A.

Sum of Hermitian matrices is Hermitian

The sum of any two Hermitian matrices is Hermitian.

Proof

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

Inverse of Hermitian matrix is Hermitian

The inverse 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 $A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}.$

Commutative product of Hermitian matrices is Hermitian

The product of two Hermitian matrices A and B is Hermitian if and only if AB = 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 $~AB=BA.$

Consequence: If a matrix $A$ is Hermitian, then for any integer $m\geq 0,~A^{m}$ is Hermitian.

ABA is Hermitian

If two matrices $A$ and $B$ are Hermitian, then $ABA$ is also Hermitian.

Proof

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

v^HAv is real for complex v

If a matrix $A$ is Hermitian, then for any complex-valued vector $\mathbf {v} ,$ the product $\mathbf {v} ^{\mathsf {H}}\!A\mathbf {v}$ is real. Indeed, $\mathbf {v} ^{\mathsf {H}}\!A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}\!A\mathbf {v} \right)^{\mathsf {H}}.$ This is especially important in quantum physics, where Hermitian matrices are operators that measure properties of a system (e.g., total spin), which have to be real.

Complex Hermitian matrices form vector space over ℝ

The Hermitian complex n × n matrices do not form a vector space over the complex numbers, $\mathbb {C} ,$ since the identity matrix $I_{n}$ is Hermitian, but $\mathbf {i} I_{n}$ is not. However, the complex Hermitian matrices do form a vector space over the real numbers, $\mathbb {R} .$ In the ⁠$2n^{2}$⁠-dimensional vector space of complex n × n matrices over $\mathbb {R} ,$ the complex Hermitian matrices form a subspace of dimension $n^{2}.$
E_jk denoting the n × n matrix with a 1 in the j,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 set of matrices of the form

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

where $\mathbf {i}$ denotes the imaginary unit ${\Big (}\mathbf {i} ={\sqrt {-1}}{\Big )}.$

An example is that the four Pauli matrices form a complete basis for the vector space of all complex 2 × 2 Hermitian matrices over $\mathbb {R} .$

Eigendecomposition

If $n$ orthonormal eigenvectors $\mathbf {u} _{1},\dots ,\mathbf {u} _{n}$ of a Hermitian matrix $A$ are chosen and written as the columns of a matrix $U,$ then one eigendecomposition of $A$ is

$$A=U\!\varLambda U^{\mathsf {H}},$$

where $U$ is a unitary matrix—i.e., $UU^{\mathsf {H}}=I=U^{\mathsf {H}}U,$ and $\varLambda$ is a diagonal matrix with ⁠$A$⁠'s eigenvalues on ⁠$\varLambda$⁠'s diagonal. Therefore,

$$A_{ij}=\mathbf {v} _{i}\varLambda \mathbf {v} _{j}\!\!^{\mathsf {H}}=\sum _{k=1}^{k=n}\mathbf {v} _{i,k}\lambda _{k}{\overline {\mathbf {v} _{j,k}}},$$

where $\mathbf {v} _{i}$ is the row vector corresponding to the ⁠$i$⁠-th row of $U,~$ $\mathbf {v} _{i,k}$ is the ⁠$k$⁠-th coordinate of $\mathbf {v} _{i},$ and $\lambda _{k}$ is the ⁠$k$⁠-th eigenvalue on ⁠$\varLambda$⁠'s diagonal.

Singular values

The singular values of a matrix are the absolute values of its eigenvalues.

Since a Hermitian matrix $A$ has an eigendecomposition $A=U\!\varLambda U^{\mathsf {H}},$ where $U$ is a unitary matrix (its columns are orthonormal vectors—see above, and its rows are also orthonormal vectors), a singular value decomposition of $A$ is

$$A=U|\varLambda |\operatorname {sgn}(\!\varLambda )U^{\mathsf {H}},$$

where $|\varLambda |$ and $\operatorname {sgn}(\!\varLambda )$ are diagonal matrices respectively containing the absolute values $|\lambda _{i}|$ and signs $\operatorname {sgn}(\lambda _{i})$ of ⁠$A$⁠'s eigenvalues. The matrix $\operatorname {sgn}(\!\varLambda )U^{\mathsf {H}}$ is unitary, since the rows (not the columns) of $U^{\mathsf {H}}$ are only getting multiplied by $\pm 1.$ The matrix $|\varLambda |$ contains the singular values of $A,$ namely, the absolute values of ⁠$A$⁠'s eigenvalues.

Real determinant

The determinant of a Hermitian matrix is real.

Proof

For any complex matrix $A,$

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

Therefore,

$A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}.$

Other proof

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

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 $\left(A-A^{\mathsf {H}}\right)$ is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B. This is known as the Toeplitz decomposition of C. $$C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right).$$

Rayleigh quotient

Main article: Rayleigh quotient

For a complex (or real) matrix $M$ and a non-zero complex (or real) vector $\mathbf {v} ,$ the Rayleigh quotient, $R(M,\mathbf {v} ),$ is defined as the scalar:

$$R(M,\mathbf {v} ):={\frac {\mathbf {v} ^{\mathsf {H}}\!M\mathbf {v} }{\mathbf {v} ^{\mathsf {H}}\mathbf {v} }}.$$

For a real vector $\mathbf {v} ,$ the conjugate transpose $\mathbf {v} ^{\mathsf {H}}$ reduces to the usual transpose $\mathbf {v} ^{\mathsf {T}}.$

For any non-zero complex (or real) scalar $c,$

$$R(M,c\mathbf {v} )=R(M,\mathbf {v} ).$$

If $M$ is Hermitian (or symmetric real), then for any non-zero complex vector $\mathbf {v} ,~$ $R(M,\mathbf {v} )$ is real.

For a real matrix, the condition of being Hermitian reduces to that of being symmetric.

It can be shown that, for a given Hermitian (or symmetric real) matrix $M,$ the Rayleigh quotient reaches its minimum value $\lambda _{\min }$ (the smallest eigenvalue of $M$) when $\mathbf {v} =\mathbf {v} _{\min }$ (the corresponding eigenvector). Similarly, $R(M,\mathbf {v} )\leq \lambda _{\max }$ 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 a fixed 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 to M associates the Rayleigh quotient R(M, v) for a fixed v and M varying through the algebra would be referred to as a "vector state" of the algebra.

See also

• Complex symmetric matrix – Matrix equal to its transposePages displaying short descriptions of redirect targets
• Haynsworth inertia additivity formula – Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix
• Hermitian form – Generalization of complex inner productsPages displaying short descriptions of redirect targets
• Normal matrix – Matrix that commutes with its conjugate transpose
• Schur–Horn theorem – Characterizes the diagonal of a Hermitian matrix with given eigenvalues
• Self-adjoint operator – Linear operator equal to its own adjoint
• Skew-Hermitian matrix – Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix)
• Unitary matrix – Complex matrix whose conjugate transpose equals its inverse
• Vector space – Algebraic structure in linear algebra

External links

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