Hermitian Matrix
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A Hermitian Matrix is a symmetric complex matrix that is equal to its own conjugate transpose.
- AKA: Self-Adjoint Matrix.
- …
- Example(s):
- a Positive-Definite Matrix, such as ...
- See: Real Symmetric Matrix, Symmetric Matrix, Eigenvalues And Eigenvectors, Complex Number, Complex Conjugate.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Hermitian_matrix Retrieved:2015-2-16.
- 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 -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : :[math]\displaystyle{ a_{ij} = \overline{a_{ji}} }[/math] or [math]\displaystyle{ A = \overline {A^\text{T}} }[/math], in matrix form.
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix [math]\displaystyle{ A }[/math] is denoted by [math]\displaystyle{ A^\dagger }[/math], then the Hermitian property can be written concisely as :[math]\displaystyle{ A = A^\dagger. }[/math]
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.
- 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 -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : :[math]\displaystyle{ a_{ij} = \overline{a_{ji}} }[/math] or [math]\displaystyle{ A = \overline {A^\text{T}} }[/math], in matrix form.