Bounding the Norm of Matrix Powers
Daniel Ammon Dowler · ScholarsArchive (Brigham Young University) · 2013
In this paper I investigate properties of square complex matrices of the form Ak, where A is also a complex matrix, and k is a nonnegative integer. I look at several ways of representing Ak. In particular, I present an identity expressing the kth power of the Schur form T of A in terms of the elements of T, which can be used together with the Schur decomposition to provide an expression of Ak. I also explain bounds on the norm of Ak, including some based on the element-based expression of Tk. Finally, I provide a detailed exposition of the most current form of the Kreiss Matrix Theorem.