Section 32.2 Motivation
We continue on our quest to answer Question 28.1.1. If we are happy with triangular block form, then we have succeeded: every matrix is similar (over \(\C\)) to one in triangular block form. But could we make a triangular block form similar to some even simpler form?
Philosophy of inquiry: break the problem apart.
If a problem is made up of readily-identifiable smaller and/or simpler parts, concentrate on each individual part separately, if possible.
A matrix in triangular block form is immediately recognizable as being made out of parts that are both smaller and simpler: the blocks themselves. But we can break each block into parts as well.
Each block in triangular block form is in scalar-triangular form, and a scalar-triangular form matrix can be decomposed as a sum
\begin{equation*}
\lambda I + N
\end{equation*}
of a scalar matrix \(\lambda I\) and a nilpotent matrix \(N\text{.}\) As we realized in Discovery 32.1, attempting to simplify each block in triangular form is equivalent to simplifying each nilpotent part of the scalar-triangular blocks. However, just as we did when we considered scalar-triangular form as an initial special case before analyzing general triangular block form, we will consider first consider a special case of nilpotent matrix.
Philosophy of inquiry: analyze the “cusp” case.
A “cusp” is a boundary between two different kinds of behaviour, in analogy with a cusp point on the graph of a function.
The graph of an unspecified function appears on a set of axes. Initially the graph is increasing and concave up, halfway through the diagram it abruptly changes to decreasing and concave up, creating a sharp “corner” in the graph at the point of transition between these two behaviours. A vertical dashed line is drawn from this sharp point to the horizontal axis, and that location on the horizontal axis is labelled \(c\text{.}\)
A cusp is often where interesting things happen or where special patterns are revealed.
In Chapter 24, we learned that a singular matrix must have \(\lambda = 0\) as an eigenvalue (Theorem 24.6.3). So in the triangular form for a singular matrix, one of the blocks will have zeros down the diagonal, with as many diagonal zeros as the algebraic multiplicity of eigenvalue \(\lambda = 0\) in the characteristic polynomial. We often think of a singular matrix as being “like zero”, since \(0\) is the only number that does not have an inverse. But an \(n \times n\) zero matrix has eigenvalue \(\lambda = 0\) with maximum algebraic multiplicity \(m_\lambda = n\text{.}\) So, in a way, the algebraic multiplicity of \(\lambda = 0\) for a singular matrix measures how “close” to being zero that matrix is.
A Venn diagram illustrating a partition of the set of \(n \times n\) matrices based on the multiplicity of eigenvalue 0. A large rectangle, taller than it is wide, is labelled as representing the collection of all \(n \times n\) matrices. A horizontal line through the middle of the rectangle divides it into two smaller rectangles, with the upper half labelled as representing the collection of invertible \(n \times n\) matrices and the lower half labelled as representing the collection of singular \(n \times n\) matrices. Inside the upper half is the additional label “matrices with \(\lambda = 0\) of multiplicity \(0\text{.}\)”
The lower half of the large rectangle is further subdivided into four rectangular cells:
-
The first cell is labelled as representing the collection of singular \(n \times n\) matrices with eigenvalue \(\lambda = 0\) of multiplicity \(1\text{.}\)
-
The second cell is labelled as representing the collection of singular \(n \times n\) matrices with eigenvalue \(\lambda = 0\) of multiplicity \(2\text{.}\)
-
The third cell contains three dots in vertical line, suggesting that there are more un-visualized cells in which the pattern continues.
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The last cell is labelled as representing the collection of singular \(n \times n\) matrices with eigenvalue \(\lambda = 0\) of multiplicity \(n\text{.}\)
But the zero matrix is not the sole example of a matrix with eigenvalue \(\lambda = 0\) of maximum algebraic multiplicity — every nilpotent matrix does as well (Statement 2 of Theorem 31.5.3). So in one sense, nonzero nilpotent is as close as you can get to being zero without actually being zero.
However, we can further partition the set of nilpotent matrices of a particular size by degree of nilpotency — the lowest positive exponent which takes the nilpotent matrix to zero. The zero matrix stands alone in this partition, as it is the only matrix with degree of nilpotency equal to \(1\) (since it already is zero).
A Venn diagram illustrating a partition of the set of nilpotent \(n \times n\) matrices based on the degree of nilpotency. On the left side of the diagram is a large rectangle labelled as representing the collection of singular \(n \times n\) matrices. This rectangle is subdivided into four rectangular cells:
-
The first cell is labelled as representing the collection of singular \(n \times n\) matrices with eigenvalue \(\lambda = 0\) of multiplicity \(1\text{.}\)
-
The second cell is labelled as representing the collection of singular \(n \times n\) matrices with eigenvalue \(\lambda = 0\) of multiplicity \(2\text{.}\)
-
The third cell contains three dots in vertical line, suggesting that there are more un-visualized cells in which the pattern continues.
-
The last cell is outlined in bold and labelled as representing the collection of singular \(n \times n\) matrices with eigenvalue \(\lambda = 0\) of multiplicity \(n\text{.}\)
On the right side of the diagram is a second large rectangle representing an exploded view of the multiplicity-\(n\) cell of the first large rectangle, and also labelled as representing the collection of nilpotent \(n \times n\) matrices. This second large rectangle is further subdivided into five rectangular cells:
-
The first cell is labelled as representing the collection of those nilpotent \(n \times n\) matrices \(A\) satisfying \(A^n = \zerovec\) but \(A^{n - 1} \neq \zerovec\text{.}\)
-
The second cell is labelled as representing the collection of those nilpotent \(n \times n\) matrices \(A\) satisfying \(A^{n - 1} = \zerovec\) but \(A^{n - 2} \neq \zerovec\text{.}\)
-
The third cell contains three dots in vertical line, suggesting that there are more un-visualized cells in which the pattern continues.
-
The second-to-last cell is labelled as representing the collection of those nilpotent \(n \times n\) matrices \(A\) satisfying \(A^2 = \zerovec\) but \(A \neq \zerovec\text{.}\)
-
The last cell is labelled as representing the collection of consisting solely of the \(n \times n\) zero matrix.
The “cusp” case we will consider is at the top of this partition: those nilpotent matrices with degree of nilpotency \(n\text{.}\) These matrices are “close” to zero in the sense that they are nilpotent (and so have eigenvalue \(\lambda = 0\) with maximum algebraic multiplicity). But of all nilpotent matrices they are as far from zero as possible, since it takes the maximum exponent \(n\) for them to reveal their nilpotency. They are on the cusp between being nilpotent and being singular nonnilpotent.
We have already encountered a prototypical example of this sort of nilpotent matrix that is “on the cusp”: the elementary nilpotent form matrices of Discovery 32.3. Our task in this chapter (that we have already begun in Discovery guide 32.1) is to determine the pattern of similarity that emerges when a nilpotent matrix is similar to an elementary nilpotent one.
