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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.