I am kinda lost here. All I did until now was finding eigenvalues and vectors for a matrix but as far as I can understand the question it asks me to find the eigenvalues of a Linear Transformation?
Let V be the vector space of all real polynomials p(x) of degree ≤ n. Define
T : V → V, T (p) = q, q(t) = p(t+1). Determine the eigenvalues and the eigenvectors of T.
I am just looking for the general idea to start the problem. Any help is appreciated.
Best Answer
Hint: You can represent your transformation $T$ as a matrix $A$ w.r.t. to some choice of a basis of polynomials. Find the eigenvectors of $A$ and translate them back to polynomials.
From $Ax = \lambda x$ for some $\lambda \in \mathbb C$ and a nonzero coordinate vector $x$ of a nozero polynomial $p$, you can deduce that the coordinate vector of $Tp$ will be $\lambda x$, i.e. the coordinate vector of $\lambda p$, implying $$Tp = \lambda p,$$ which is the eigenvalue equation for the eigenvalue $\lambda$ of the transformation $T$ with the eigenfunction $p$.
Alternatively, you could also solve the question which polynomials satisfy $\lambda p(t) = p(t+1)$ for some $\lambda$.