16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-28A simple method to find eigenvectors for 2 × 2 matricesLet( )a bA =c dbe a 2 × 2 matrix with characteristic polynomialand let r 1 be a root of z(r).We wish to find a vector v =z(r) = r 2 − (a + b)r + ad − bc,(v1v 2)such thatAv = r 1 v or (A − r 1 I)v = 0with I equal to the 2 × 2 identity matrix.That is, we wish to solve the system of equations(a − r 1 )v 1 + bv 2 = 0cv 1 + (d − r 1 )v 2 = 0This is a homogeneous system of linear equations, and, since it has asolution, the two equations must be multiples of each other. Thus, we onlyneed to solve the first equation.Case 1: b ≠ 0Since b ≠ 0, we can let v 1 = 1 and get v 2 = r 1−ato solve the equation.bHence, the vectorv =(1r 1 −abis an eigenvector for r 1 .Note that this works whether the root is real or complex. In the complexcase, we get an associated complex eigenvector–there is no associated realeigenvector.If the two roots of z(r) are r 1 , r 2 with both real and distinct, then thesame formula works for each of them.)