Section 34.1 Motivation
Finally, we reach the end of our journey to answering Question 28.1.1. We know we can make every (complex) matrix similar to one in triangular block form (Theorem 30.5.1). Each block in triangular block form is a scalar-triangular block that can be broken up into the sum of a scalar part and a nilpotent part. Using the patterns from Discovery 32.1 and Subsection 32.4.1, we saw that can make a triangular block form matrix similar to something simpler by building a second transition matrix in block diagonal form out of blocks that would make the nilpotent part of each scalar-triangular block simpler. And “simpler” for a nilpotent matrix means triangular-block nilpotent form. Putting triangular-block nilpotent form back together with the scalar part of each block in triangular block form, we get something called Jordan normal form.