EIGENVALUE

eigenvalue, eigenvalue of a matrix, eigenvalue of a square matrix, characteristic root of a square matrix

(noun) (mathematics) any number such that a given square matrix minus that number times the identity matrix has a zero determinant

Source: WordNet® 3.1


Etymology

Noun

eigenvalue (plural eigenvalues)

(linear algebra) A scalar, \(\lambda\), such that there exists a non-zero vector \(x\) (a corresponding eigenvector) for which the image of \(x\) under a given linear operator \(\mathrm{A}\) is equal to the image of \(x\) under multiplication by \(\lambda\); i.e. \(\mathrm{A} x = \lambda x\).

The eigenvalues \(\lambda\) of a square transformation matrix \(\mathrm{M}\) may be found by solving \(\det(\mathrm{M} - \lambda\mathrm{I}) = 0\).

Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by \(\mathrm{M} x = \lambda x\) for some right eigenvector \(x\). Left eigenvalues, characterised by \(y \mathrm{M} = y\lambda\) also exist with associated left eigenvectors \(y\). (In consequence of the equations, left eigenvectors are row vectors, while right eigenvectors are column vectors.) The convention of right eigenvector as "standard" is fundamentally an arbitrary choice.

Synonyms

• (scalar multiplier of an eigenvector): characteristic root, characteristic value, eigenroot, latent value, proper value

Source: Wiktionary



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Word of the Day

7 July 2024

SECURE

(adjective) free from danger or risk; “secure from harm”; “his fortune was secure”; “made a secure place for himself in his field”


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Coffee Trivia

Contrary to popular belief, coffee beans are not technically beans. They are referred to as such because of their resemblance to legumes. A coffee bean is a seed of the Coffea plant and the source for coffee. It is the pit inside the red or purple fruit, often referred to as a cherry. Just like ordinary cherries, the coffee fruit is also a so-called stone fruit.

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