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.

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