Thread: eigenvalues and basis of F(R,R)

So the question says to find all real eigenvalues and for each of these real eigenvalues find a real eigenfunction. V is the linear space of all functions
R->R.
a) T(f(x))=2f(x) now I can already see the only eigenvalue is 2 and any function with real coefficients works so f(x)=sinx is one such eigenfunction.

b) T(f(x))=f(-x)
Now the way I would usually approach this is I would find the matrix A=[T] with respect to the standard ordered basis of V and then use the characteristic polynomial f(t)=det(A-tI) BUT I can't figure out what the standard ordered basis would be. a basis for all polynomials would be (1, x, x^2,...) but in V, the space of ALL functions from R to R, functions like sinx are included so how can I find the matrix representation of T?

I know, for example, -1 and 1 are eigenvalues because T(sinx)=-sinx and T(cosx)=cosx but what about some function g(x)={1, when x>or=0 and g(x)={4, when x<0 ? wouldnt T(g(x))=4g(x) so can't any number be an eigenvalue of T?

Also T(h(x))=-(1/3)x if h(x)=abs(x)+2x so -1/3 is also an eigenvalue for T...so an any number be an eigenvalue? how do I show that?

Originally Posted by 234578

So the question says to find all real eigenvalues and for each of these real eigenvalues find a real eigenfunction. V is the linear space of all functions
R->R.
a) T(f(x))=2f(x) now I can already see the only eigenvalue is 2 and any function with real coefficients works so f(x)=sinx is one such eigenfunction.

b) T(f(x))=f(-x)
Now the way I would usually approach this is I would find the matrix A=[T] with respect to the standard ordered basis of V and then use the characteristic polynomial f(t)=det(A-tI) BUT I can't figure out what the standard ordered basis would be. a basis for all polynomials would be (1, x, x^2,...) but in V, the space of ALL functions from R to R, functions like sinx are included so how can I find the matrix representation of T?

I know, for example, -1 and 1 are eigenvalues because T(sinx)=-sinx and T(cosx)=cosx but what about some function g(x)={1, when x>or=0 and g(x)={4, when x<0 ? wouldnt T(g(x))=4g(x) so can't any number be an eigenvalue of T?

Also T(h(x))=-(1/3)x if h(x)=abs(x)+2x so -1/3 is also an eigenvalue for T...so an any number be an eigenvalue? how do I show that?

For the first one you're supposed to find functions such that where and which will give you so you're right.

For the second one, remember that matrix representations only work for finite dim. spaces and yours isn't. Now you want and so so for you have all even functions as eigenvectors and for all odd functions.

Originally Posted by Jose27

For the first one you're supposed to find functions such that where and which will give you so you're right.

For the second one, remember that matrix representations only work for finite dim. spaces and yours isn't. Now you want and so so for you have all even functions as eigenvectors and for all odd functions.

thanks for the reply, could you possibly clarify some things I didnt understand? how did you get f(x)=af(-x)=a^2f(x) ? is it because we know analytically that the function must be either even or odd?

Originally Posted by 234578

thanks for the reply, could you possibly clarify some things I didnt understand? how did you get f(x)=af(-x)=a^2f(x) ? is it because we know analytically that the function must be either even or odd?

Apply your operator twice, or use a change of variables and apply the property that f is an eigenvector. Or its easy to see that your whole space is the direct sum of the subspace of even functions and the subspace of odd functions