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eigenvalue (Definition)

Let $V$ be a vector space over $k$ and $T$ a linear operator on $V$. An eigenvalue for $T$ is an scalar $\lambda$ (that is, an element of $k$) such that $T(z)=\lambda z$ for some nonzero vector $z\in V$. Is that case, we also say that $z$ is an eigenvector of $T$.

This can also be expressed as follows: $\lambda$ is an eigenvalue for $T$ if the kernel of $A-\lambda I$ is non trivial.

A linear operator can have several eigenvalues (but no more than the dimension of the space). Eigenvectors corresponding to different eigenvalues are linearly independent.



"eigenvalue" is owned by drini. [ full author list (2) | owner history (2) ]
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See Also: linear transformation, scalar, Euclidean vector, kernel, dimension (vector space)

Also defines:  eigenvector

Attachments:
eigenspace (Definition) by CWoo
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Cross-references: linearly independent, dimension, eigenvalues, kernel, vector, scalar, linear operator, vector space
There are 14 references to this entry.

This is version 5 of eigenvalue, born on 2003-10-15, modified 2004-06-03.
Object id is 5106, canonical name is EigenvalueOfALinearOperator.
Accessed 12404 times total.

Classification:
AMS MSC: 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)
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