If we have the linear operator: $T\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$ = $\begin{pmatrix} 2c & a+c \\ b-2c & d \\ \end{pmatrix}$ How would I find the eigenvalues and eigenvectors?
What I was trying to do was making the $$\det\left(\lambda I- \begin{pmatrix} 2c & a+c \\ b-2c & d \\ \end{pmatrix}\right)$$ But somehow I feel this is wrong, how can I do it?
There is no use of the matrix structure, meaning we can decide on an order as long as we stay consistent: $$ T \left( \begin{array}{c} a \\ b \\ c \\ d \end{array} \right) = \left( \begin{array}{c} 2c \\ a+c \\ b-2c \\ d \end{array} \right) $$ In turn, there is a 4 by 4 matrix that accomplishes this. Find it and its eigenvalues
This is the same idea as for polynomials of degree no larger than some $k.$ It is a matter of being careful about picking a basis (not necessarily writing it out) and writing in that basis.