Hermitian 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:

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

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

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

In this section, the conjugate transpose of matrix ${\displaystyle A}$ is denoted as ${\displaystyle A^{H}}$, the transpose of matrix ${\displaystyle A}$ is denoted as ${\displaystyle A^{T}}$ and conjugate of matrix ${\displaystyle A}$ is denoted as ${\displaystyle A^{*}}$.

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