Consider the non-trivial eigenvalue equation Ax = λx. Dividing through by λ gives [A/λ]x = x. Can it be said that any vector x that solves this equation is a fixed-point in the vector space? And because the output of the operator [A/λ] is equal to the vector it acts on, this equation is a self-referential statement?
It depends on what do you understand by "self-referential statement" ! So you must give a definition regarding the notion called "self-referential statement" !
If E is a vector space and A:E->E is a linear operator from E to E, then λ is an eigenvalue of A if we can find a notnull vector x in E such that Ax = λx. Such vector x is called eigenvector of A relatively to the eigenvalue λ.
We can denote the set of all eigenvectors of A relatively to eigenvalue λ by SA(λ). If E has a topological structure compatible with his structure of vector space( for example, if E is a normed space) and if A is additionaly continuous on E, then SA(λ) is a closed subspace in E.
Clearly, if λ is notnull, then we can equivalently write the equation Ax = λx as
[(1/λ)A]x = x. Therefore the problem of finding all x eigenvectors of A relatively to λ is reduced to the problem of finding all fixed points of linear operator (1/λ)A.
So such x satisfying the equation [(1/λ)A]x = x is a fixed point of operator
(1/λ)A, not a "fixed-point in the vector space", this last statement having not a clear meaning in mathematics.
It's important to observe that 0 is always a solution of equation [(1/λ)A]x = x( 0 is always a fixed point of operator (1/λ)A) because (1/λ)A is a linear operator. Consequently, if the equation [(1/λ)A]x = x has only the trivial solution(x=0), then λ is not an eigenvalue of A.
Thank you Dinu.
I think what you are saying is this: I should say that x is a fixed-point of the operator, rather than saying its a fixed-point in the vector space. Is that correct?
Yes, dear Steve, it's correct! We can speak only on " fixed point(fixed points) of an operator( of a function)" ! "Fixed point in a vector space" has not a meaning in mathematics.
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