A self-adjoint matrix, also known as a Hermitian matrix in the context of complex numbers, is a square matrix that is equal to its own conjugate transpose. This means that if ( A ) is a self-adjoint matrix, it holds true that ( A = A^ ), where ( A^ ) denotes the conjugate transpose of ( A ).
Frequency of Use: Self-adjoint matrices are commonly encountered in higher mathematics, especially in linear algebra, functional analysis, and quantum mechanics. Their usage is more prevalent in written academic and technical contexts than in everyday oral communication.
Example Sentences:
- A self-adjoint matrix has real eigenvalues.
Оперный матрица имеет действительные собственные значения.
The properties of a self-adjoint matrix are essential in quantum mechanics.
Свойства самосопряженной матрицы важны в квантовой механике.
To determine if a matrix is self-adjoint, calculate its transpose and compare it to the original matrix.
Чтобы определить, является ли матрица самосопряженной, вычислите её транспонированную матрицу и сравните с исходной матрицей.
The term self-adjoint matrix is not typically found in idiomatic expressions, as it is a specialized term in mathematics. However, the concepts associated with it, such as eigenvalues and orthogonality, create foundational contexts in various mathematical idioms.
Example Sentences with Related Concepts:
- Eigenvalues of a self-adjoint matrix provide insight into its geometric properties.
Собственные значения самосопряженной матрицы дают представление о её геометрических свойствах.
The term self-adjoint combines self (indicating that the subject is equal to itself) and adjoint (from Latin "adiunctus," meaning "joined or connected"), which refers to a mathematical operation where a matrix is paired or related to another matrix via transposition and conjugation.
Synonyms: - Hermitian matrix (in complex number contexts) - Symmetric matrix (for real-valued matrices)
Antonyms: - Non-self-adjoint matrix - Non-Hermitian matrix
This structured response provides a comprehensive overview of the term "self-adjoint matrix," detailing its use, significance in mathematics, and related concepts.