Eigenvalues of Matrices of Low Rank

While it can be difficult, or even impossible, to find exact eigenvalues of an n X n matrix in general, we will illustrate a simple method that works when the rank of the matrix is small. This technique can be used for student discovery in a linear algebra class; an instructor can assign a sequence of exercises requiring students to solve special cases, make conjectures about generalizations, and then prove their conjectures. The eigenvalues of a matrix A are the zeroes of its characteristic polynomial, det(A/A), which can be written as P(A) =A"-C 11 _ 1 A"1 +c 11 _ 2 A"2 ... +(-1)"c0 . (1)