The Cayley-Hamilton Theorem, Pt. 1

Every square matrix satisfies its characteristic equation.

Our central question of this series boils down to finding which powers of a matrix are equal. We therefore need ways to talk about the powers of matrices. In the past few articles, we used the eigendecomposition of the matrix to convert the problem of powers of matrices to powers of elements in a finite field.

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This method worked 80% of the time, but we also found cycle lengths and delays that could not be explained with an eigendecomposition. The eigendecomposition method is breaking down. We need to take a step back and analyze the powers of matrices directly. This analysis will lead us to the Cayley-Hamilton Theorem.

Prerequisites

While this is part of a series, you don’t need to know anything from the previous articles except:

• An Abelian group is a set of objects that acts like the integers under addition.
• A commutative unital ring is a set of objects you can add, subtract, and multiply like the integers.
• A field is a set of objects you can add, subtract, multiply, and divide exactly like the rational numbers.