Definition:Hermitian Matrix

Definition

Let $\mathbf A$ be a square matrix over $\C$.

$\mathbf A$ is Hermitian if and only if:

$\mathbf A = \mathbf A^\dagger$

where $\mathbf A^\dagger$ is the Hermitian conjugate of $\mathbf A$.

Also known as

A Hermitian matrix is also known just as a Hermitian.

Examples

Arbitrary Example

This is an example of a Hermitian matrix:

$\begin {pmatrix} 1 & i \\ -i & 1 \end {pmatrix}$

Also see

• Definition:Hermitian Operator
• Definition:Symmetric Matrix
• Definition:Unitary Matrix
• Definition:Anti-Hermitian Matrix

• Results about Hermitian matrices can be found here.

Source of Name

This entry was named for Charles Hermite.

Historical Note

The Hermitian matrix was invented by Charles Hermite to solve certain problems in number theory.

However, it turned out to be crucial in the formulation by Werner Karl Heisenberg of his model of quantum mechanics.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hermitian conjugate
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Hermitian matrix
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hermitian conjugate
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Hermitian matrix
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Hermitian