Suppose that a 2x2 real matrix has the form . For which values of the numbers a and b will B be diagonalizable by a real matrix? Explain.
Remember that when you diagonalize a matrix B, the diagonal values of the resulting diagonalized matrix will be the eigenvalues of B, and that a matrix P such that $\displaystyle P^{-1}BP$ is diagonal will have as it's columns the eigenvectors corresponding to those eigenvalues. Thus, we require that B have real eigenvectors. As our matrix is real, real eigenvalues are a necessary condition for real eigenvectors.
Note that $\displaystyle \det(I\lambda-B)=0$ gives $\displaystyle (\lambda-a)^2+b^2=0$.
This should point you in the right direction.
--Kevin C.