How to find eigenvalue of infinite dimensional matrix?
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One simple example with a special matrix, which has somehow "a continuum" as eigenvalue...
Consider some function f(x)=K+ax+bx2+cx3+... having a nonzero radius of convergence. Then think of the infinite matrix of the form
From the properties of finite matrices we would expect, that K is an eigenvalue. But consider a type of an infinite vector
V(x)=[1,x,x2,x3,x4,…]
with a scalar parameter x from the range of convergence, then
V(x)⋅F=f(x)⋅V(x)
This means also: any vector V(x) is an eigenvector of the matrix F and corresponds to the eigenvalue f(x). If now f(x) is entire, for instance the exponential function f(x)=exp(x), then any value from the complex plane (except 0 because exp(x) is never 0) "is an eigenvalue" of F contradicting the "naive" extrapolation from the finite truncation of the matrix ...
Hope this helps you ☺️☺️✌️✌️❤️❤️
Consider some function f(x)=K+ax+bx2+cx3+... having a nonzero radius of convergence. Then think of the infinite matrix of the form
From the properties of finite matrices we would expect, that K is an eigenvalue. But consider a type of an infinite vector
V(x)=[1,x,x2,x3,x4,…]
with a scalar parameter x from the range of convergence, then
V(x)⋅F=f(x)⋅V(x)
This means also: any vector V(x) is an eigenvector of the matrix F and corresponds to the eigenvalue f(x). If now f(x) is entire, for instance the exponential function f(x)=exp(x), then any value from the complex plane (except 0 because exp(x) is never 0) "is an eigenvalue" of F contradicting the "naive" extrapolation from the finite truncation of the matrix ...
Hope this helps you ☺️☺️✌️✌️❤️❤️
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