Eigenvalues and Eigenfunctions of an Operator

Question

Suppose I am given two orthonormal vectors $v_1$ and $v_2$ which constitute a base of a two-dimensional space. Suppose also that an operator $T$ defined on this space is satisfying some equations of the form: $Tv_1=av_2+bv_2$ and $Tv_2=cv_1+dv_2$. How do I calculate the eigenvalues and the orthonormal eigenfunctions of $T$?

Answer 1

In that basis the matrix corresponding to $T$ is $\begin{pmatrix}a&c\\b&d\end{pmatrix}$. To see this, I encourage you to see the definition of a matrix of an operator in a given basis. The eigenvalues and eigenfunctions in this basis can be found by the usual means for matrices. In the end you just need to interpret $\begin{pmatrix}x\\y\end{pmatrix}$ as $xv_1+yv_2$.

Answer 2

This is a finite dimensional question, so you may write the operator in matrix form (using the basis $v_1,v_2$) as $$ \begin{bmatrix}a&c\\b&d \end{bmatrix} $$ then solve for the roots of characteristic polynomial to find eigenvalues $\lambda$ (there may be none, one, or two depending on what field you are over!) solving $$ \lambda^2-(a+d)\lambda+\det(T)=0 $$ If you get an eigenvalue $\lambda$, you find eigenvectors by examining $$ \ker\left(\begin{bmatrix}a-\lambda&c\\b&d-\lambda \end{bmatrix}\right) $$