Eigenvalues and Eigenvectors for a linear transformation T. Please help

Question

Consider $ T : R[x]_{≤2} → R[x]_{≤2} $ defined by $$ (T f)(x) = \int_{-1}^1 (x − y)^2 f(y) dy − 2f(0)x^2 $$ for $ f ∈ R[x]_{≤2}. $ Find all eigenvalues and eigenvectors for T.

Answer 1

Write down a matrix representing $\;T\;$, say wrt to the "usual" basis $\;\{1,x,x^2\}\;$, for example:

$$(Tx)(x):=\int_{-1}^1(x-y)^2y\,dy-2\cdot0x^2=\int_{-1}^1\left(xy-2xy^2+y^3\right)\,dy=\left.-\frac23xy^3\right|_{-1}^1=-\frac43x$$

and since $\;-\cfrac43x=\color{red}0\cdot1+\color{red}{\left(-\cfrac43\right)}x+\color{red}0\cdot x^2\;$ , your matrix's second column is $\;\begin{pmatrix}0\\-\frac43\\0\end{pmatrix}\;$

Well, after you have your matrix, just evaluate the characteristic polynomial, then its roots and...voila.