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The theory of anti-eigenvalues is a [[Spectral theory|spectral theory]] based upon the turning angles of a matrix or operator  $  A $. 
(See [[Eigen value|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|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 [[#References|[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|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 [[#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  $  Ax = b $; 
see [[#References|[a5]]], [[#References|[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 [[#References|[a4]]]. Even the basic Richardson method  $  x _ {k + 1 }  = x _ {k} + \alpha ( b - Ax _ {k} ) $(
cf. also [[Richardson extrapolation|Richardson extrapolation]]) may be seen to have optimal convergence rate  $  \rho _ {\textrm{ opt  }  } =  \sin  A $. 
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>