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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 i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:

${\displaystyle a_{ij}={\overline {a_{ji}}}\,.}$

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

${\displaystyle A=A^{\dagger }\,.}$

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

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:

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

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).

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, and the finite-dimensional spectral theorem applies. It 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 are real, and that A has n linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of C consisting of n eigenvectors of A.

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 A and B is Hermitian if and only if AB = BA. Thus A is Hermitian if A is Hermitian and n is an integer.

The Hermitian complex n-by-n 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 2n-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n. If E_jk denotes the n-by-n matrix with a 1 in the j,k position and zeros elsewhere, a basis can be described as follows:

${\displaystyle \;E_{jj}}$ for ${\displaystyle 1\leq j\leq n}$ (n matrices)

together with the set of matrices of the form

${\displaystyle \;E_{jk}+E_{kj}}$ for ${\displaystyle 1\leq j<k\leq n}$ (⁠n − n/2⁠ matrices)

and the matrices

${\displaystyle \;i(E_{jk}-E_{kj})}$ for ${\displaystyle 1\leq j<k\leq n}$ (⁠n − n/2⁠ matrices)

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

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

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

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

Additional facts related to Hermitian matrices include:

• The sum of a square matrix and its conjugate transpose ${\displaystyle (C+C^{\dagger })}$ is Hermitian.
• The difference of a square matrix and its conjugate transpose ${\displaystyle (C-C^{\dagger })}$ is skew-Hermitian (also called antihermitian).
  • This implies that 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:

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

• The determinant of a Hermitian matrix is real:

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

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

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

See also

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

References

1. Frankel, Theodore (2004). The geometry of physics: an introduction. Cambridge University Press. p. 652. ISBN 0-521-53927-7.
2. Physics 125 Course Notes at California Institute of Technology

External links

• "Hermitian matrix", Encyclopedia of Mathematics, EMS Press, 2001
• Visualizing Hermitian Matrix as An Ellipse with Dr. Geo, by Chao-Kuei Hung from Shu-Te University, gives a more geometric explanation.
• "Hermitian Matrices". MathPages.com.

• Matrices