Chapter 6 Jordan normal form

3cm If you think about things the way someone else does then you will never understand it as well as if you think about it your own way.

– Manjul Bhargava (1974 - )

Fields medallist

Not every matrix is diagonalizable; for example is not diagonalizable for any field $F$. So it is not always possible to replace a matrix $A$ with a diagonal matrix which is similar to it (recall the definition of similar matrices from 4.53). But, the main purpose of this final Chapter (where we usually assume $F=ℂ$) is to find a similar matrix which is as close as possible to being diagonal. This will be called the Jordan normal form of the matrix. This method is used in MATH318 for finding solutions to certain systems of differential equations, and also in MATH319 for finding the exponential of a square matrix.

First we will look at the fundamental, and somewhat surprising, Cayley-Hamilton theorem, and follow that up with a procedure for determining the minimal polynomial.