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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?

Hint

Theorem 26.5.1.

(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*}

(See (\(\star\)) in Subsection 31.3.2.)

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?

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 \(A\text{.}\)

Hint

Corollary 26.5.6.

(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 = \underline{\hspace{2.272727272727273em}}, \amp A \uvec{p}_2 \amp = \underline{\hspace{2.272727272727273em}}, \amp A \uvec{p}_3 \amp = \underline{\hspace{2.272727272727273em}}, \amp A \uvec{p}_4 \amp = \underline{\hspace{2.272727272727273em}}\text{.} \end{align*}
(b)

Reinterpret the pattern you described in Task a:

\begin{align*} A \uvec{p}_1 \amp = \underline{\hspace{2.272727272727273em}}, \amp A^2 \uvec{p}_1 \amp = \underline{\hspace{2.272727272727273em}}, \amp A^3 \uvec{p}_1 \amp = \underline{\hspace{2.272727272727273em}}, \amp A^4 \uvec{p}_1 \amp = \underline{\hspace{2.272727272727273em}}\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 find 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 = \left[\begin{array}{rrr} 0 \amp 2 \amp 1\\ -1 \amp 2 \amp 1\\ 2 \amp -4 \amp -2 \end{array}\right] \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.)