Convergence of matrix powers
Stephen H. Friedberg, Arnold J. Insel · International Journal of Mathematical Education in Science and Technology · 1992
We provide an elementary and short proof of the following important result: Let A be an n x nreal or complex matrix. Then {Am} converges if and only the following two conditions hold: (a) if A is an eigenvalue of A, then either A = 1 or A lies in the open unit disk of the complex plane; and (b) if 1 is an eigenvalue of A, then its algebraic multiplicity equals its geometric multiplicity. Unlike the standard proofs of this result, our proof requires no knowledge of Jordan canonical forms. In addition, the proof provides students with an example of the interplay between matrix techniques and vector space theory.