A primer on eigenfunctions

Question

This is the basics on how to solve them. Suppose we have $$ \left( \begin {array}{cc} a & 0\\ 0 & b \\ \end{array} \right).$$ In this case I can tell the eigen vectors are $$ \left( \begin {array}{c} a \\ 0 \end {array} \right)$$ and $$ \left( \begin {array}{c} 0 \\ b \end {array} \right),$$ right? But how can I solve something like $$ \left( \begin {array}{cc} a & b\\ c & d \\ \end{array} \right)?$$ I'm writing this question because I never got the idea behind the procedure, so that's what I'm looking for.

Answer 1

In a general case, you first compute the eigenvalues of the matrix by solving the equation:

$|A-\lambda I|=0 $

Afterwards, you substitute the values of the eigenvalues in the matrix $A-\lambda I$ and calculate the column basis for each case. Those basis for the column vectors are the eigenvectors of the matrix.