Anti-eigenvalue
The theory of anti-eigenvalues is a spectral theory based upon the turning angles of a matrix or operator . (See Eigen value for the spectral theory of stretchings, rather than turnings, of a matrix or operator.)
For a strongly accretive operator , i.e.,
,
, the first anti-eigenvalue
is defined by
![]() | (a1) |
From (a1) one has immediately the notion of the angle : the largest angle through which
may turn a vector. Any corresponding vector
which is turned by that angle is called a first anti-eigenvector. It turns out that, in general, the first anti-eigenvectors come in pairs. Two important early results were the minmax theorem and the Euler equation.
Minmax theorem.
For any strongly accretive bounded operator on a Hilbert space
,
![]() | (a2) |
![]() |
Using the minmax theorem, the right-hand side of (a2) is seen to define
![]() | (a3) |
in such a way that . This implies an operator trigonometry (see [a1]).
Euler equation.
For any strongly accretive bounded operator on a Hilbert space
, the Euler equation for the anti-eigenvalue functional
in (a1) is
![]() | (a4) |
![]() |
When is a normal operator, (a4) is satisfied not only by the first anti-eigenvectors of
, but by all eigenvectors of
. Therefore the Euler equation may be viewed as a significant extension of the Rayleigh–Ritz theory for the variational characterization of eigenvalues of a self-adjoint or normal operator
. The eigenvectors maximize the variational quotient (a1). The anti-eigenvectors minimize it. See [a2], [a3].
The theory of anti-eigenvalues has been applied recently (from 1990 onward) to gradient and iterative methods for the solution of linear systems ; see [a5], [a6]. For example, the Kantorovich convergence rate for steepest descent,
![]() |
where denotes the
-inner-product error
, becomes
![]() |
Thus, the Kantorovich error rate is trigonometric. Similar trigonometric convergence bounds hold for conjugate gradient and related more sophisticated algorithms [a4]. Even the basic Richardson method (cf. also Richardson extrapolation) may be seen to have optimal convergence rate
. For further information, see [a5], [a6].
References
[a1] | K. Gustafson, "Operator trigonometry" Linear Multilinear Alg. , 37 (1994) pp. 139–159 |
[a2] | K. Gustafson, "Antieigenvalues" Linear Alg. & Its Appl. , 208/209 (1994) pp. 437–454 |
[a3] | K. Gustafson, "Matrix trigonometry" Linear Alg. & Its Appl. , 217 (1995) pp. 117–140 |
[a4] | K. Gustafson, "Operator trigonometry of iterative methods" Numerical Linear Alg. Appl. , to appear (1997) |
[a5] | K. Gustafson, "Lectures on computational fluid dynamics, mathematical physics, and linear algebra" , Kaigai & World Sci. (1996/7) |
[a6] | K. Gustafson, D. Rao, "Numerical range" , Springer (1997) |
Anti-eigenvalue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-eigenvalue&oldid=45192