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Pronunciation

enPR: Ä«Ê¹gÉ™n'vÄƒlyoÍžo, IPA: /ËˆaÉªÉ¡É™nËŒvÃ¦ljuË/

Origin

- + value

Full definition of eigenvalue

Noun

eigenvalue

(plural eigenvalues)

(linear algebra) A scalar, $\lambda\!$ , such that there exists a vector $x$ (the corresponding eigenvector) for which the image of

x under a given linear operator

\rm A\!

is equal to the image of

x

under multiplication by

\lambda

i.e.

{\rm A} x = \lambda x\!

''The eigenvalues $\lambda\!$ of a square transformation matrix

\rm M\! may be found by solving

\det({\rm M} - \lambda {\rm I}) = 0\!

.

Usage notes

When unqualified, as in the above example, eigenvalue conventionally refers to a right eigenvalue, characterised by ${\rm M} x = \lambda x\!$ for some right eigenvector $x\!$

. Left eigenvalues, charactarised by $y {\rm M} = y \lambda\!$ also exist with associated left eigenvectors $y\!$

. For commutative operators, the left eigenvalues and right eigenvalues will be the same, and are referred to as eigenvalues with no qualifier.

Eigenvalue