Hermitian matrix

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose—that is, the element in the Template:Mvar-th row and Template:Mvar-th column is equal to the complex conjugate of the element in the Template:Mvar-th row and Template:Mvar-th column, for all indices Template:Mvar and Template:Mvar:

$a_{ij}={\overline {a_{ji}}}$ or $A={\overline {A^{\text{T}}}}$, in matrix form.

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

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

$A=A^{\dagger }.$

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

Examples

See the following example:

${\begin{bmatrix}2&2+i&4\\2-i&3&i\\4&-i&1\\\end{bmatrix}}$

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

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

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

Properties

• The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of a Hermitian matrix.

• Every Hermitian matrix is a normal matrix.

• 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 Template:Mvar are real, and that Template:Mvar has Template:Mvar linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of Cⁿ consisting of Template:Mvar eigenvectors of Template:Mvar.

• The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices Template:Mvar and Template:Mvar is Hermitian if and only if AB = BA. Thus Aⁿ is Hermitian if Template:Mvar is Hermitian and Template:Mvar is an integer.

• For an arbitrary complex valued vector Template:Mvar the product $v^{\dagger }Av$ is real because of $v^{\dagger }Av=(v^{\dagger }Av)^{\dagger }$. 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.

• The Hermitian complex Template:Mvar-by-Template:Mvar matrices do not form a vector space over the complex numbers, since the identity matrix I_n is Hermitian, but i I_n is not. However the complex Hermitian matrices do form a vector space over the real numbers R. In the 2n²-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n². If E_jk denotes the Template:Mvar-by-Template:Mvar matrix with a 1 in the j,k position and zeros elsewhere, a basis can be described as follows:

$\;E_{jj}$ for $1\leq j\leq n$ (Template:Mvar matrices)

together with the set of matrices of the form

$\;E_{jk}+E_{kj}$ for $1\leq j<k\leq n$ ({{ safesubst:#invoke:Unsubst||$B=n² − n/2}} matrices)

and the matrices

$\;i(E_{jk}-E_{kj})$ for $1\leq j<k\leq n$ ({{ safesubst:#invoke:Unsubst||$B=n² − n/2}} matrices)

where $i$ denotes the complex number ${\sqrt {-1}}$, known as the imaginary unit.

• If Template:Mvar orthonormal eigenvectors $u_{1},\dots ,u_{n}$ of a Hermitian matrix are chosen and written as the columns of the matrix Template:Mvar, then one eigendecomposition of Template:Mvar is $A=U\Lambda U^{\dagger }$ where $UU^{\dagger }=I=U^{\dagger }U$ and therefore

$A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\dagger }$,

where $\lambda _{j}$ are the eigenvalues on the diagonal of the diagonal matrix $\;\Lambda$.

Further properties

{{safesubst:#invoke:anchor|main}}Additional facts related to Hermitian matrices include:

• The sum of a square matrix and its conjugate transpose $(C+C^{\dagger })$ is Hermitian.
• The difference of a square matrix and its conjugate transpose $(C-C^{\dagger })$ is skew-Hermitian (also called antihermitian). This implies that commutator of two Hermitian matrices is skew-Hermitian.
• An arbitrary square matrix Template:Mvar can be written as the sum of a Hermitian matrix Template:Mvar and a skew-Hermitian matrix Template:Mvar:

$C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}(C+C^{\dagger })\quad {\mbox{and}}\quad B={\frac {1}{2}}(C-C^{\dagger }).$

• The determinant of a Hermitian matrix is real:

Proof: $\det(A)=\det(A^{\mathrm {T} })\quad \Rightarrow \quad \det(A^{\dagger })=\det(A)^{*}$

Therefore if $A=A^{\dagger }\quad \Rightarrow \quad \det(A)=\det(A)^{*}.$

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

Rayleigh quotient

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See also

• Skew-Hermitian matrix (anti-Hermitian matrix)
• Haynsworth inertia additivity formula
• Hermitian form
• Self-adjoint operator
• Unitary matrix

External links

• {{#invoke:citation/CS1|citation

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• Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
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