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