Linear and multilinear algebra; matrix theory Definition

Hermitian matrix

Builds on Conjugate transpose Square matrix

Definition

Let $A = (a_{ij})$ be an $n \times n$ complex square matrix. The matrix $A$ is Hermitian when $A^* = A$ , where $A^*$ is the conjugate transpose of $A$ . Equivalently, $a_{ij} = \overline{a_{ji}}$ for all $i$ and $j$ .

Remarks

The diagonal entries of a Hermitian matrix are real. A real matrix is Hermitian if and only if it is symmetric. The eigenvalues of a Hermitian matrix are real, and $\C^n$ has an orthonormal basis of eigenvectors, by the spectral theorem for Hermitian matrices. The matrix $\begin{pmatrix} 2 & 1 - i \\ 1 + i & 3 \end{pmatrix}$ is Hermitian.

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Matrix definition
Square matrix definition
Transpose definition
Conjugate transpose definition
Hermitian matrix definition

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36 pages build on this one.

Tag	`0EKZCX4K3P`
URI	`m2://0EKZCX4K3P`
Kind	definition
Writing	W1 stated
MSC	15B57
Defines	Hermitian matrix
Uses	Conjugate transpose, Square matrix

Hermitian matrix

Definition

Remarks

Reading path

Where this leads

Further reading

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