diagonalizable

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Definition

Adjective: 1. Capable of being transformed into a diagonal matrix: In linear algebra, a square matrix is described as diagonalizable if it can be converted into a diagonal matrix through a similarity transformation. This typically involves finding a basis of eigenvectors for the matrix.

Examples of Usage
Advanced Usage
  • Diagonalizable over a field: A matrix is said to be diagonalizable over a specific field (like the real numbers ℝ or complex numbers ℂ) if the necessary eigenvectors and eigenvalues exist within that field. A matrix may not be diagonalizable over ℝ but can be diagonalizable over ℂ.
    • The rotation matrix is not diagonalizable over the real numbers, but it is diagonalizable over the complex numbers.
Variants and Related Words
  • Diagonalize (verb): The process of converting a matrix into diagonal form.
    • We can diagonalize the matrix using its eigenvectors.
  • Diagonalizability (noun): The property or condition of being diagonalizable.
    • The diagonalizability of the operator simplifies the analysis.
  • Diagonal (adjective/noun): Relating to the straight line joining opposite corners; in matrices, a diagonal matrix has non-zero entries only on its main diagonal.
Synonyms
  • Similar to a diagonal matrix: This is a more technical, equivalent description.
  • Possessing a complete eigenbasis: This describes the underlying condition for diagonalizability.
Related Concepts (Not Phrasal Verbs/Idioms)
  • Eigenvalue: A scalar associated with a linear transformation.
  • Eigenvector: A non-zero vector that changes only by a scalar factor when a linear transformation is applied.
  • Similarity Transformation: An operation of the form P⁻¹AP, which preserves eigenvalues.
  • Spectral Theorem: A theorem stating conditions under which a matrix is diagonalizable, particularly for normal matrices (e.g., symmetric or Hermitian matrices).
Adjective
  1. capable of being transformed into a diagonal matrix