diagonalise
Definition
Verb: 1. To transform a matrix into a diagonal matrix: In linear algebra, to perform a specific mathematical operation on a square matrix to find an equivalent diagonal matrix. This process typically involves finding a matrix of eigenvectors and its inverse.
Usage
The verb "diagonalise" is used in the context of mathematics, specifically linear algebra. It describes the action performed on a matrix. * The subject is the person or process performing the transformation. * The object is the matrix being transformed.
Examples
- Verb:
- The goal of the exercise was to diagonalise the given 3x3 matrix.
- Not every square matrix can be diagonalised; some can only be put into Jordan form.
- Once you diagonalise the matrix, calculating its powers becomes trivial.
Advanced Usage
- "to be diagonalisable" (adjective): Describes a matrix for which the operation of diagonalisation is possible.
- A symmetric matrix is always diagonalisable.
- "diagonalisation" (noun): The process or result of diagonalising.
- The diagonalisation of the system's matrix simplified the differential equations.
Variants and Related Words
- Diagonalize (verb): The preferred spelling in American English.
- Diagonalisable (adj): The British English spelling of the adjective.
- Diagonalizable (adj): The American English spelling of the adjective.
- Diagonal (adj/n): The adjective describing something arranged in a sloping line, or the noun for the line itself. This is the root word from which "diagonalise" is derived in the matrix context, referring to the entries on the main diagonal.
Synonyms
- Transform to diagonal form: A more descriptive phrase with the same meaning.
Related Concepts (Not Phrasal Verbs or Idioms)
- Eigenvector: A non-zero vector that changes only by a scalar factor when a linear transformation is applied. Essential for diagonalisation.
- Eigenvalue: The scalar factor by which its corresponding eigenvector is scaled.
- Similarity Transformation: The operation , which, when is a matrix of eigenvectors, yields a diagonal matrix.
Verb
-
transform a matrix to a diagonal matrix
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