View source for Anti-eigenvalue

The theory of anti-eigenvalues is a [[Spectral theory|spectral theory]] based upon the turning angles of a matrix or operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106301.png" />. (See [[Eigen value|Eigen value]] for the spectral theory of stretchings, rather than turnings, of a matrix or operator.)

For a strongly accretive operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106302.png" />, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106304.png" />, the first anti-eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106305.png" /> is defined by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106306.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

From (a1) one has immediately the notion of the angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106307.png" />: the largest angle through which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106308.png" /> may turn a vector. Any corresponding vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a1106309.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063010.png" /> on a [[Hilbert space|Hilbert space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063011.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063013.png" /></td> </tr></table>

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

in such a way that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063015.png" />. This implies an operator trigonometry (see [[#References|[a1]]]).

==Euler equation.==
For any strongly accretive bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063016.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063017.png" />, the Euler equation for the anti-eigenvalue functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063018.png" /> in (a1) is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063019.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063020.png" /></td> </tr></table>

When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063021.png" /> is a [[Normal operator|normal operator]], (a4) is satisfied not only by the first anti-eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063022.png" />, but by all eigenvectors of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063023.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063024.png" />. The eigenvectors maximize the variational quotient (a1). The anti-eigenvectors minimize it. See [[#References|[a2]]], [[#References|[a3]]].

The theory of anti-eigenvalues has been applied recently (from 1990 onward) to gradient and iterative methods for the solution of linear systems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063025.png" />; see [[#References|[a5]]], [[#References|[a6]]]. For example, the Kantorovich convergence rate for steepest descent,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063026.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063027.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063028.png" />-inner-product error <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063029.png" />, becomes

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063030.png" /></td> </tr></table>

Thus, the Kantorovich error rate is trigonometric. Similar trigonometric convergence bounds hold for conjugate gradient and related more sophisticated algorithms [[#References|[a4]]]. Even the basic Richardson method <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063031.png" /> (cf. also [[Richardson extrapolation|Richardson extrapolation]]) may be seen to have optimal convergence rate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110630/a11063032.png" />. For further information, see [[#References|[a5]]], [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Gustafson,   "Operator trigonometry"  ''Linear Multilinear Alg.'' , '''37'''  (1994)  pp. 139–159</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Gustafson,   "Antieigenvalues"  ''Linear Alg. &amp; Its Appl.'' , '''208/209'''  (1994)  pp. 437–454</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Gustafson,   "Matrix trigonometry"  ''Linear Alg. &amp; Its Appl.'' , '''217'''  (1995)  pp. 117–140</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Gustafson,   "Operator trigonometry of iterative methods"  ''Numerical Linear Alg. Appl.'' , '''to appear'''  (1997)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  K. Gustafson,   "Lectures on computational fluid dynamics, mathematical physics, and linear algebra" , Kaigai &amp; World Sci.  (1996/7)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Gustafson,   D. Rao,   "Numerical range" , Springer  (1997)</TD></TR></table>