For matrices with symmetry over the real number field, see Symmetric matrix.
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix 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:
or in matrix form:
Hermitian matrices can be understood as the complex extension of real symmetric matrices.
If the conjugate transpose of a matrix is denoted by then the Hermitian property can be written concisely as
Hermitian matrices are named after Charles Hermite,[1] who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are although in quantum mechanics, typically means the complex conjugate only, and not the conjugate transpose.
^Archibald, Tom (2010-12-31), Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), "VI.47 Charles Hermite", The Princeton Companion to Mathematics, Princeton University Press, pp. 773–773, doi:10.1515/9781400830398.773a, ISBN 978-1-4008-3039-8, retrieved 2023-11-15
In mathematics, a Hermitianmatrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element...
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overline) is called a Hermitianmatrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if A T = A ¯ . {\displaystyle...
Dieudonné determinant Moore, E. H. (1922), "On the determinant of an hermitianmatrix with quaternionic elements. Definition and elementary properties with...
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symmetric matrix and more generally a Hermitianmatrix, and a unitary matrix, is never defective; more generally, a normal matrix (which includes Hermitian and...
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{\displaystyle A\mathbf {x} =\mathbf {b} } . Applicable to: square, hermitian, positive definite matrix A {\displaystyle A} Decomposition: A = U ∗ U {\displaystyle...
{\displaystyle A} is a Hermitianmatrix and ‖ ⋅ ‖ {\displaystyle \|\cdot \|} is the Euclidean norm. This is because any HermitianMatrix is diagonalizable...
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semi-definite Hermitianmatrix ( U {\displaystyle U} is an orthogonal matrix and P {\displaystyle P} is a positive semi-definite symmetric matrix in the real...
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