# Anti-eigenvalue

The theory of anti-eigenvalues is a spectral theory based upon the turning angles of a matrix or operator $A$. (See Eigen value for the spectral theory of stretchings, rather than turnings, of a matrix or operator.)

For a strongly accretive operator $A$, i.e., ${ \mathop{\rm Re} } \langle {Ax,x } \rangle \geq m \| x \| ^ {2}$, $m > 0$, the first anti-eigenvalue $\mu$ is defined by

$$\tag{a1 } \mu \equiv \cos A = \inf _ {x \in D ( A ) } { \frac{ { \mathop{\rm Re} } \left \langle {Ax, x } \right \rangle }{\left \| {Ax } \right \| \left \| x \right \| } } .$$

From (a1) one has immediately the notion of the angle $\phi ( A )$: the largest angle through which $A$ may turn a vector. Any corresponding vector $x$ 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 $A$ on a Hilbert space $X$,

$$\tag{a2 } \sup _ {\left \| x \right \| = 1 } \inf _ {- \infty < \epsilon < \infty } \left \| {( \epsilon A - I ) x } \right \| ^ {2} =$$

$$= \inf _ {- \infty < \epsilon < \infty } \sup _ {\left \| x \right \| = 1 } \left \| {( \epsilon A - I ) x } \right \| ^ {2} .$$

Using the minmax theorem, the right-hand side of (a2) is seen to define

$$\tag{a3 } \nu = \sin A = \inf _ {\epsilon > 0 } \left \| {\epsilon A - I } \right \|$$

in such a way that $\cos ^ {2} A + \sin ^ {2} A = 1$. This implies an operator trigonometry (see [a1]).

## Euler equation.

For any strongly accretive bounded operator $A$ on a Hilbert space $X$, the Euler equation for the anti-eigenvalue functional $\mu$ in (a1) is

$$\tag{a4 } 2 \left \| {Ax } \right \| ^ {2} \left \| x \right \| ^ {2} ( { \mathop{\rm Re} } A ) x - \left \| x \right \| ^ {2} { \mathop{\rm Re} } \left \langle {Ax,x } \right \rangle A ^ {*} Ax +$$

$$- \left \| {Ax } \right \| ^ {2} { \mathop{\rm Re} } \left \langle {Ax,x } \right \rangle x = 0.$$

When $A$ is a normal operator, (a4) is satisfied not only by the first anti-eigenvectors of $A$, but by all eigenvectors of $A$. 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 $A$. 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 $Ax = b$; see [a5], [a6]. For example, the Kantorovich convergence rate for steepest descent,

$$E _ {A} ( x _ {k + 1 } ) \leq \left ( 1 -4 \lambda _ {1} \lambda _ {n} ( \lambda _ {1} + \lambda _ {n} ) ^ {-2 } \right ) E _ {A} ( x _ {k} ) ,$$

where $E _ {A}$ denotes the $A$- inner-product error $\langle {( x - x ^ {*} ) , A ( x - x ^ {*} ) } \rangle$, becomes

$$E _ {A} ( x _ {k + 1 } ) \leq ( \sin ^ {2} A ) E _ {A} ( x _ {k} ) .$$

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 $x _ {k + 1 } = x _ {k} + \alpha ( b - Ax _ {k} )$( cf. also Richardson extrapolation) may be seen to have optimal convergence rate $\rho _ {\textrm{ opt } } = \sin A$. For further information, see [a5], [a6].

How to Cite This Entry:
Anti-eigenvalue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Anti-eigenvalue&oldid=45192
This article was adapted from an original article by K. Gustafson (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article