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

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

Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by $\backslash (\backslash mathrm\{M\}\; x\; =\; \backslash lambda\; x\backslash )$ for some right eigenvector $\backslash (x\backslash )$. Left eigenvalues, characterised by $\backslash (y\; \backslash mathrm\{M\}\; =\; y\backslash lambda\backslash )$ also exist with associated left eigenvectors $\backslash (y\backslash )$. (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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