eigenvalue

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English[edit]

Etymology[edit]

eigen- +‎ value

Pronunciation[edit]

  • enPR: īʹgən'vălyo͞o, IPA(key): /ˈaɪɡənˌvæljuː/
  • (file)

Noun[edit]

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[edit]

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.

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