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askIITians Faculty 747 Points
6 years ago
In mathematics, aHermitian matrix(orself-adjoint matrix) is asquare matrixwithcomplexentries that is equal to its ownconjugate transpose—that is, the element in thei-th row andj-th column is equal to thecomplex conjugateof the element in thej-th row andi-th column, for all indicesiandj:

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

Hermitian matrices can be understood as the complex extension of realsymmetric matrices.

If theconjugate transposeof a matrix [A] is denoted by [A^\dagger] , then the Hermitian property can be written concisely as

[A = A^\dagger.]

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

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