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Hermitian Matrix


A square matrix is called Hermitian if it is self-adjoint. Therefore, a Hermitian matrix A=(a_(ij)) is defined as one for which

 A=A^(H),
(1)

where A^(H) denotes the conjugate transpose. This is equivalent to the condition

 a_(ij)=a^__(ji),
(2)

where z^_ denotes the complex conjugate. As a result of this definition, the diagonal elements a_(ii) of a Hermitian matrix are real numbers (since a_(ii)=a^__(ii)), while other elements may be complex.

Examples of 2×2 Hermitian matrices include

 [1 -i; i 1],[2 -i; i 1]
(3)

and the Pauli matrices

sigma_1=[0 1; 1 0]
(4)
sigma_2=[0 -i; i 0]
(5)
sigma_3=[1 0; 0 -1].
(6)

Examples of 3×3 Hermitian matrices include

 [-1 1-2i 0; 1+2i 0 -i; 0 i 1],[1 1+i 2i; 1-i 5 -3; -2i -3 0].
(7)

An integer or real matrix is Hermitian iff it is symmetric.

A matrix m can be tested to see if it is Hermitian in the Wolfram Language using HermitianMatrixQ[m].

Hermitian matrices have real eigenvalues whose eigenvectors form a unitary basis. For real matrices, Hermitian is the same as symmetric.

Any matrix C which is not Hermitian can be expressed as the sum of a Hermitian matrix and a antihermitian matrix using

 C=1/2(C+C^(H))+1/2(C-C^(H)).
(8)

Let U be a unitary matrix and A be a Hermitian matrix. Then the adjoint of a similarity transformation is

(UAU^(-1))^(H)=[(UA)(U^(-1))]^(H)
(9)
=(U^(-1))^(H)(UA)^(H)
(10)
=(U^H)^(H)(A^(H)U^(H))
(11)
=UAU^(H)
(12)
=UAU^(-1).
(13)

The specific matrix

H(x,y,z)=[z x-iy; x+iy -z]
(14)
=xP_1+yP_2+zP_3,
(15)

where P_i are Pauli matrices, is sometimes called "the" Hermitian matrix.


See also

Adjoint, Antihermitian Matrix, Conjugate Transpose, Hermitian Operator, Hermitian Part, Normal Matrix, Pauli Matrices, Symmetric Matrix

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References

Arfken, G. "Hermitian Matrices, Unitary Matrices." §4.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 209-217, 1985.Ayres, F. Jr. Schaum's Outline of Theory and Problems of Matrices. New York: Schaum, pp. 13 and 117-118, 1962.

Referenced on Wolfram|Alpha

Hermitian Matrix

Cite this as:

Weisstein, Eric W. "Hermitian Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HermitianMatrix.html

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