HERMITIAN MATRIX Meaning and
Definition
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A Hermitian matrix refers to a square matrix that is equal to its own conjugate transpose. In simpler terms, it is a matrix whose complex entries are symmetrical across the main diagonal and whose conjugate transpose is the same as the original matrix. Hermitian matrices are exclusively defined for complex numbers and are a generalization of real symmetric matrices.
In detail, for a square matrix A with entries a_ij, it is said to be Hermitian if a_ij = a_ji* for all i and j, where a_ji* represents the complex conjugate of a_ji. This implies that the entries on the main diagonal are always real numbers. Thus, a Hermitian matrix has a characteristic property of symmetry, but also incorporates complex numbers.
Importantly, Hermitian matrices play a fundamental role in various areas of mathematics and physics. They have several significant properties, such as all their eigenvalues being real numbers and their eigenvectors being orthogonal. Due to their symmetric nature, they are often used in solving systems of linear equations, performing spectral analysis, and studying quantum mechanics.
The term "Hermitian" is derived from the name of Charles Hermite, a renowned French mathematician who made notable contributions to the theory of complex numbers and transcendentals. His work laid the foundation for the development and understanding of Hermitian matrices.
Common Misspellings for HERMITIAN MATRIX
- germitian matrix
- bermitian matrix
- nermitian matrix
- jermitian matrix
- uermitian matrix
- yermitian matrix
- hwrmitian matrix
- hsrmitian matrix
- hdrmitian matrix
- hrrmitian matrix
- h4rmitian matrix
- h3rmitian matrix
- heemitian matrix
- hedmitian matrix
- hefmitian matrix
- hetmitian matrix
- he5mitian matrix
- he4mitian matrix
- hernitian matrix
- herkitian matrix
Etymology of HERMITIAN MATRIX
The term "Hermitian matrix" is derived from the name of the French mathematician Charles Hermite. Charles Hermite was a 19th-century mathematician known for his work in number theory, algebra, and mathematical analysis.
The concept of a Hermitian matrix is closely related to the concept of a symmetric matrix. A matrix is symmetric if it is equal to its transpose, meaning that the elements of the matrix along its main diagonal remain the same, while the elements across the main diagonal are swapped.
A Hermitian matrix is the complex analogue of a symmetric matrix. In addition to being equal to its transpose, a Hermitian matrix also satisfies the condition that the conjugate of each element is equal to the corresponding entry in the matrix. This means that a Hermitian matrix has complex conjugate symmetries.
