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Eigenvalue vs Eigenspace - What's the difference?

eigenvalue | eigenspace |

In linear algebra|lang=en terms the difference between eigenvalue and eigenspace

is that eigenvalue is (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; ie {\rm a} x = \lambda x\! while eigenspace is (linear algebra) a set of the eigenvectors associated with a particular eigenvalue, together with the zero vector.

As nouns the difference between eigenvalue and eigenspace

is that eigenvalue is (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; ie {\rm a} x = \lambda x\! while eigenspace is (linear algebra) a set of the eigenvectors associated with a particular eigenvalue, together with the zero vector.

eigenvalue

English

Noun

(en noun)
  • (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.

    Synonyms

    * characteristic root * characteristic value * eigenroot * latent value * proper value

    Related terms

    * eigen decomposition * eigenface * eigenfunction * eigenmode * eigenstate * eigenvector

    See also

    * ("eigenvalue" on Wikipedia) * Mathworld article on eigenvalues

    eigenspace

    English

    Noun

    (en noun)
  • (linear algebra) A set of the eigenvectors associated with a particular eigenvalue, together with the zero vector.