# eigenvalue

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## English

### Pronunciation

• enPR: īʹgən'vălyo͞o, IPA(key): /ˈaɪɡənˌvæljuː/
•  Audio (US) Sorry, your browser either has JavaScript disabled or does not have any supported player. You can download the clip or download a player to play the clip in your browser. (file)

### Noun

eigenvalue (plural eigenvalues)

1. (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.