In mathematics, the Fredholm alternative, named after Ivar Fredholm, is one of Fredholm's theorems and is a result in Fredholm theory. It may be expressed in several ways, as a theorem of linear algebra, a theorem of integral equations, or as a theorem on Fredholm operators. Part of the result states that a non-zero complex number in the spectrum of a compact operator is an eigenvalue.
- For each vector v in V there is a vector u in V so that . In other words: T is surjective (and so also bijective, since V is finite-dimensional).
A more elementary formulation, in terms of matrices, is as follows. Given an m×n matrix A and a m×1 column vector b, exactly one of the following must hold:
- Either: A x = b has a solution x
- Or: AT y = 0 has a solution y with yTb ≠ 0.
In other words, A x = b has a solution if and only if for any y s.t. AT y = 0, yTb = 0 .
and the inhomogeneous equation
The Fredholm alternative states that, for any non-zero fixed complex number , either the first equation has a non-trivial solution, or the second equation has a solution for all .
A sufficient condition for this theorem to hold is for to be square integrable on the rectangle (where a and/or b may be minus or plus infinity).
Loosely speaking, the correspondence between the linear algebra version, and the integral equation version, is as follows: Let
or, in index notation,
is given by
with given by
In this language, the integral equation alternatives are seen to correspond to the linear algebra alternatives.
In more precise terms, the Fredholm alternative only applies when K is a compact operator. From Fredholm theory, smooth integral kernels are compact operators. The Fredholm alternative may be restated in the following form: a nonzero is either an eigenvalue of K, or it lies in the domain of the resolvent
- E.I. Fredholm, "Sur une classe d'equations fonctionnelles", Acta Math., 27 (1903) pp. 365–390.
- A. G. Ramm, "A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators", American Mathematical Monthly, 108 (2001) p. 855.
- Khvedelidze, B.V. (2001), "Fredholm theorems for integral equations", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Weisstein, Eric W., "Fredholm Alternative", MathWorld.