Given:
Eigenvalues:
There are two roots: andThe eigenvector corresponding to satisfies
Taking v1y = -2, then v1x = 1, giving an eigenvector (1,-2). Normalizing this vector to length one gives:The eigenvector corresponding to satisfies
Taking v2y = 1, then v2x = 2, giving after normalization:Finally:
Check:
The diagonal form is what matrix A looks like in the coordinate system x',y' shown below: