Section 33.1 Motivation
Just as scalar-triangular form was a warm-up for triangular block form, elementary nilpotent form was a warm-up for what we will call triangular-block nilpotent form:
\begin{equation*}
\begin{bmatrix}
N_1 \\
\amp N_2 \\
\amp \amp \ddots \\
\amp \amp \amp N_\ell
\end{bmatrix}\text{,}
\end{equation*}
where each block \(N_j\) is an elementary nilpotent matrix.
The theory behind determining a transition matrix that will take a nilpotent matrix to this new form is technically involved. While we will provide a procedure to create such a transition matrix in this chapter, we will be more concerned with just being able to determine the exact form matrix from properties of the original nilpotent matrix.
Philosophy of inquiry: gather information indirectly.
Rather than try to compute or observe information directly, use more easily obtainable information as clues to the nature of the desired, more complicated information.