Geometrically, one thinks of a vector whose direction is unchanged by the action of , but whose magnitude is multiplied by .
(2) The linear mapping
i.e. , has no inverse.
(3) is not injective.
(4) is not surjective.
(5) , i.e. .
But if is of infinite dimension, (5) has no meaning and the conditions (2) and (4) are not equivalent to (1). A scalar satisfying (2) (called a spectral value of ) need not be an eigenvalue. Consider for example the complex vector space of all sequences of complex numbers with the obvious operations, and the map given by
Zero is a spectral value of , but clearly not an eigenvalue.
Now suppose again that is of finite dimension, say . The function
is a polynomial of degree over in the variable , called the characteristic polynomial of the endomorphism . (Note that some writers define the characteristic polynomial as rather than , but the two have the same zeros.)
If is or any other algebraically closed field, or if and is odd, then has at least one zero, meaning that has at least one eigenvalue. In no case does have more than eigenvalues.
Although we didn’t need to do so here, one can compute the coefficients of by introducing a basis of and the corresponding matrix for . Unfortunately, computing determinants and finding roots of polynomials of degree are computationally messy procedures for even moderately large , so for most practical purposes variations on this naive scheme are needed. See the eigenvalue problem for more information.
If but the coefficients of are real (and in particular if has a basis for which the matrix of has only real entries), then the non-real eigenvalues of appear in conjugate pairs. For example, if and, for some basis, has the matrix
then , with the two zeros .
Eigenvalues are of relatively little importance in connection with an infinite-dimensional vector space, unless that space is endowed with some additional structure, typically that of a Banach space or Hilbert space. But in those cases the notion is of great value in physics, engineering, and mathematics proper. Look for “spectral theory” for more on that subject.
|Date of creation||2013-03-22 12:11:52|
|Last modified on||2013-03-22 12:11:52|
|Last modified by||Koro (127)|