diagonalisation

Học thuật
Thân thiện
diagonalisation

A mathematician writes a diagonalisation of a matrix on a whiteboard.

Definition
  1. Noun:
    • The process of converting a square matrix into a diagonal matrix: This involves finding a basis in which the matrix representation is diagonal, meaning all non-zero elements are on the main diagonal from the top-left to the bottom-right corner. This is a key concept in linear algebra.
Usage Examples
  • Noun:
    • The diagonalisation of the matrix revealed its eigenvalues.
    • Performing diagonalisation simplifies many matrix computations.
Advanced Usage
  • "Diagonalisation by a similarity transformation": Refers to the standard method where a matrix A is diagonalised by finding an invertible matrix P such that P⁻¹AP is diagonal.
    • The lecture covered diagonalisation by a similarity transformation for symmetric matrices.
  • "Unitary diagonalisation": A specific type of diagonalisation applicable to normal matrices (e.g., Hermitian or unitary matrices) using a unitary transformation.
    • For a Hermitian matrix, unitary diagonalisation is always possible.
Variants and Related Words
  • Diagonalization (noun): The American English spelling of "diagonalisation".
  • Diagonalise (verb): To convert a matrix into diagonal form.
    • We need to diagonalise this operator to solve the equation.
  • Diagonalizable (adjective): Describing a matrix that can be converted into a diagonal form.
    • Not every square matrix is diagonalizable.
Synonyms
  • Diagonal transformation: A process resulting in a diagonal matrix.
  • Spectrum decomposition: A related concept, especially for normal operators, where the matrix is decomposed based on its eigenvalues (spectrum).
Related Phrases
  • "To bring to diagonal form": An alternative phrasing meaning to diagonalise.
    • The goal of the algorithm is to bring the matrix to diagonal form.
  • "Eigendecomposition": A decomposition of a matrix into its eigenvectors and eigenvalues, which is closely related to and often equivalent to diagonalisation for diagonalizable matrices.
    • Eigendecomposition is a fundamental application of diagonalisation.
diagonalisation

A mathematician writes a diagonalisation of a matrix on a whiteboard.

Noun
  1. changing a square matrix to diagonal form (with all non-zero elements on the principal diagonal)
    • the diagonalization of a normal matrix by a unitary transformation

Từ đồng nghĩa

Từ gần giống