eigenvalue

Let $V$ be a vector space over a field $k$, and let $A$ be anendomorphism of $V$ (meaning a linear mapping of $V$ into itself).A scalar $\lambda \in k$ is said to be aneigenvalue of $A$ if there is a nonzero $x\in V$ for which

$$Ax=\lambda x.$$(1)

Geometrically, one thinks of a vector whose direction is unchangedby the action of $A$, but whose magnitude is multiplied by $\lambda$.

If $V$ is finite dimensional, elementary linear algebra shows thatthere are several equivalent definitions of an eigenvalue:

(2) The linear mapping

$$B=\lambda I-A$$

i.e. $B:x\mapsto \lambda x-Ax$, has no inverse.

(3) $B$ is not injective.

(4) $B$ is not surjective.

(5) $det(B)=0$, i.e. $det(\lambda I-A)=0$.

But if $V$ is of infinite dimension, (5) has no meaning and theconditions (2) and (4) are not equivalent to (1).A scalar $\lambda$ satisfying (2) (called a spectral value of$A$) need not be an eigenvalue. Consider for example the complexvector space $V$ of all sequences ${(x_n)}_{n=1}^{\mathrm{\infty }}$ of complexnumbers with the obvious operations, and the map $A:V\to V$ given by

$$A(x_1,x_2,x_3,\mathrm{\dots })=(0,x_1,x_2,x_3,\mathrm{\dots }).$$

Zero is a spectral value of $A$, but clearly not an eigenvalue.

Now suppose again that $V$ is of finite dimension, say $n$.The function

$$\chi (\lambda )=det(B)$$

is a polynomial of degree $n$ over $k$ in thevariable $\lambda$, called the characteristic polynomial of theendomorphism $A$. (Note that some writers define the characteristicpolynomial as $det(A-\lambda I)$ rather than $det(\lambda I-A)$, but thetwo have the same zeros.)

If $k$ is $ℂ$ or any other algebraically closed field, or if $k=ℝ$and $n$ is odd, then $\chi$ has at least one zero, meaning that $A$has at least one eigenvalue. In no case does $A$ have more than $n$eigenvalues.

Although we didn’t need to do so here, one can compute the coefficientsof $\chi$ by introducing a basis of $V$ and the corresponding matrix for$B$. Unfortunately, computing $n\times n$ determinants and finding rootsof polynomials of degree $n$ are computationally messy proceduresfor even moderately large $n$, so for most practical purposesvariations on this naive scheme are needed. See the eigenvalueproblem for more information.

If $k=ℂ$ but the coefficients of $\chi$ are real (and in particular if$V$ has a basis for which the matrix of $A$ has only real entries), thenthe non-real eigenvalues of $A$ appear in conjugate pairs. For example,if $n=2$ and, for some basis, $A$ has the matrix

then $\chi (\lambda )={\lambda }^2+1$, with the two zeros $\pm i$.

Eigenvalues are of relatively little importance in connection withan infinite-dimensional vector space, unless that space is endowed withsome additional structure, typically that of a Banach space or Hilbert space. But in those cases the notion is of great value inphysics, engineering, and mathematics proper. Look for “spectral theory”for more on that subject.

Title	eigenvalue
Canonical name	Eigenvalue
Date of creation	2013-03-22 12:11:52
Last modified on	2013-03-22 12:11:52
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	15
Author	Koro (127)
Entry type	Definition
Classification	msc 15A18
Related topic	EigenvalueProblem
Related topic	SimilarMatrix
Related topic	Eigenvector
Related topic	SingularValueDecomposition
Defines	eigenvalue
Defines	spectral value