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Discovery guide 32.1 Discovery guide
Discovery 32.1 . Moving past triangular block form.
Suppose
\(A\) is a square matrix. Recall that
\(A\) is similar over
\(\C\) to a matrix
\(T\) in triangular block form.
(a) Suppose
\(T\) is similar to some “simpler” matrix
\(J\text{.}\) What then is the relationship between
\(A\) and this new matrix
\(J\text{?}\)
Relative to the title of this discovery activity, what is the point of this particular task?
(b)
Recall that a triangular block matrix \(T\) is a block-diagonal matrix
\begin{equation*}
T = \begin{bmatrix} T_1 \\ \amp T_2 \\ \amp \amp \ddots \\ \amp \amp \amp T_\ell \end{bmatrix} \text{,}
\end{equation*}
where each block \(T_j\) is in a particular form.
Suppose \(Q\) is a transition matrix that is also in block-diagonal form
\begin{equation*}
Q = \begin{bmatrix} Q_1 \\ \amp Q_2 \\ \amp \amp \ddots \\ \amp \amp \amp Q_\ell \end{bmatrix} \text{,}
\end{equation*}
where each block \(Q_j\) has the same size as the corresponding block \(T_j\) in \(T\text{.}\)
Express
\(\inv{Q} T Q\) in block form.
Relative to the title of this discovery activity, what is the point of this particular task?
(c)
Recall that each block \(T_j\) in a triangular block matrix \(T\) is in scalar-triangular form, and so can be decomposed as a sum of a scalar matrix \(\lambda_j I\) and a nilpotent matrix \(N_j\text{:}\)
\begin{equation*}
T_j = \lambda_j I + N_j \text{.}
\end{equation*}
Simplify expression
\(\inv{Q}_j (\lambda_j I + N_j) Q_j\text{.}\)
Relative to the title of this discovery activity, what is the point of this particular task?
Discovery 32.2 .
Verify that a matrix that is similar to a
nilpotent matrix must be nilpotent itself.
We now turn to the task suggested by
Discovery 32.1 : determining the pattern of similarity to a nilpotent matrix. As usual, we start with a special case.
Discovery 32.3 .
A matrix that is all zeros except for a line of ones down the first sub-diagonal is called
elementary nilpotent form .
Suppose \(4 \times 4\) matrix \(A\) is similar to the elementary nilpotent matrix
\begin{equation*}
N =
\begin{bmatrix}
0 \amp 0 \amp 0 \amp 0 \\
1 \amp 0 \amp 0 \amp 0 \\
0 \amp 1 \amp 0 \amp 0 \\
0 \amp 0 \amp 1 \amp 0
\end{bmatrix} \text{.}
\end{equation*}
(a) Compute the powers
\(N^2, N^3, N^4, \dotsc\) of
\(N\text{.}\) What is the pattern? What is the
degree of nilpotency of
\(N\text{?}\)
(b)
Discovery 32.2 says that
\(A\) must also be nilpotent. What is the first power of
\(A\) that is equal to the zero matrix?
(c) What is the rank of
\(A\text{?}\) What is the rank of
\(A^2\text{?}\) Continue with ranks of higher powers of
\(A\text{.}\)
(d)
Repeat all tasks of this activity for \(5 \times 5\) matrix \(A\) similar to the elementary nilpotent matrix
\begin{equation*}
N = \begin{bmatrix}
0 \amp 0 \amp 0 \amp 0 \amp 0 \\
1 \amp 0 \amp 0 \amp 0 \amp 0 \\
0 \amp 1 \amp 0 \amp 0 \amp 0 \\
0 \amp 0 \amp 1 \amp 0 \amp 0 \\
0 \amp 0 \amp 0 \amp 1 \amp 0
\end{bmatrix}\text{.}
\end{equation*}
See if you can obtain the requested properties of \(A\) without calculations, just from the patterns of your previous results for \(4 \times 4\) matrices \(A\) and \(N\text{.}\)
Discovery 32.4 .
Suppose \(A\) is similar to the elementary nilpotent matrix
\begin{equation*}
N =
\begin{bmatrix}
0 \amp 0 \amp 0 \amp 0 \\
1 \amp 0 \amp 0 \amp 0 \\
0 \amp 1 \amp 0 \amp 0 \\
0 \amp 0 \amp 1 \amp 0
\end{bmatrix}
\end{equation*}
via similarity relation \(\inv{P} A P = N\text{.}\)
(a)
Use
Pattern 26.3.1 to determine the similarity pattern in the relation
\(\inv{P} A P = N\text{:}\)
\begin{align*}
A \uvec{p}_1 \amp = \fillinmath{XXXXX},
\amp
A \uvec{p}_2 \amp = \fillinmath{XXXXX},
\amp
A \uvec{p}_3 \amp = \fillinmath{XXXXX},
\amp
A \uvec{p}_4 \amp = \fillinmath{XXXXX}\text{.}
\end{align*}
(b)
Reinterpret the pattern you described in
Task a :
\begin{align*}
A \uvec{p}_1 \amp = \fillinmath{XXXXX},
\amp
A^2 \uvec{p}_1 \amp = \fillinmath{XXXXX},
\amp
A^3 \uvec{p}_1 \amp = \fillinmath{XXXXX},
\amp
A^4 \uvec{p}_1 \amp = \fillinmath{XXXXX}\text{.}
\end{align*}
(c) Summarize the pattern of Task b.
If \(A\) is a nilpotent matrix that we suspect is similar to an elementary nilpotent matrix, to determine a suitable transition matrix \(P\) we just need to determine a suitable first column \(\uvec{p}_1\text{,}\) and then set
\begin{equation*}
P = \begin{bmatrix}
| \amp | \amp | \amp | \\
\uvec{p}_1 \amp \boxed{\phantom{XX}} \amp \boxed{\phantom{XX}} \amp \boxed{\phantom{XX}} \\
| \amp | \amp | \amp |
\end{bmatrix}\text{.}
\end{equation*}
(d) Before even considering the linear independence of the columns that you filled in for
Task c , what condition might be used to judge the “suitability” of the choice of first column
\(\uvec{p}_1\text{?}\)
Hint .
The last fill-in-the-blank for
Task b seems like it expresses a pretty specific condition about
\(\uvec{p}_1\) relative to
\(A\text{.}\) However, if
\(A\) is similar to
\(N\) then
Task b of
Discovery 32.3 showed us that we will have
\(A^4 = \zerovec\text{,}\) so that particular condition from
Task b will be true for
every vector in
\(\R^4\text{.}\)
Instead, consider the following. We know that it is possible for many different transition matrices to achieve the similarity relation between two specific, similar matrices. Suppose you already had a suitable first column to create one transition matrix
\(P_1\) so that
\(\inv{P}_1 A P_1 = N\text{.}\) Could you use any of the other columns from
\(P_1\) as the
first column in a new transition matrix
\(P_2\text{,}\) and then fill in the rest of the columns of
\(P_2\) according to your pattern in
Task c ? How would the pattern of
Task b turn out in that case?
Discovery 32.5 .
Let’s try out the similarity patterns we discovered in
Discovery 32.4 .
The following \(3 \times 3\) matrix \(A\) satisfies \(A^3 = 0\) but \(A^2 \neq 0\text{:}\)
\begin{equation*}
A = \begin{abmatrix}{rrr} 0 \amp 2 \amp 1\\ -1 \amp 2 \amp 1\\ 2 \amp -4 \amp -2 \end{abmatrix} \text{.}
\end{equation*}
(a) Verify that
\(\uvec{p}_1 = \uvec{e}_1 = (1,0,0)\) is
not a suitable choice, with “suitable” as you described in
Task d of
Discovery 32.4 .
(b) Verify that
\(\uvec{p}_1 = \uvec{e}_2 = (0,1,0)\) is a suitable choice.
(c) Now suppose
\(A\) is some
\(n \times n\) nilpotent matrix with
\(A^n = 0\) but
\(A^{n-1} \neq 0\text{.}\) Use your experience from
Task a and
Task b to devise a strategy for choosing a suitable first column
\(\uvec{p}_1\) for transition matrix
\(P\) from amongst the standard basis vectors
\(\uvec{e}_1, \uvec{e}_2, \dotsc, \uvec{e}_n \text{.}\)
Further refine your strategy based on your knowledge of the pattern of
matrix-times-standard-basis-vector . (See
Discovery 21.1 , for example.)
