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Discovery guide 31.1 Discovery guide

Discovery 31.1.
(a)

Consider the polynomial \(p(x) = x^2 - 3x + 6\text{.}\) What should \(p(A)\) mean when \(A\) is a square matrix?

(b)

Using the polynomial \(p(x)\) from Task a, verify that \(p(\inv{P} A P) = \inv{P} \bigl(p(A)\bigr) P\) will be true for all square matrices \(A\text{.}\)

Will the same be true if we use a different polynomial?

Describe the pattern of what you've discovered using the words similar and polynomial.

(c)

Determine a pattern to quickly calculate \(p(D)\) when \(D\) is diagonal that works for every polynomial \(p(x)\text{.}\)

Now repeat for block-diagonal.

Discovery 31.2.

Consider the scalar-triangular matrix

\begin{equation*} S = \begin{bmatrix} 5 \amp a \amp b \\ 0 \amp 5 \amp c \\ 0 \amp 0 \amp 5 \end{bmatrix}\text{.} \end{equation*}
(a)

What are the eigenvalues of \(S\text{?}\) What are their algebraic multiplicities? (You should be able to determine this just by inspection.)

(b)

Write the characteristic polynomial \(c_S(\lambda)\) in fully factored form. (Using your answers Task a, you should be able to do this without any calculation.)

(c)

Compute \(c_S(S)\) using your factored form of \(c_S(\lambda)\) from Task b.

What happened?

Discovery 31.3.

Let's explore the pattern of Discovery 31.2 a little more.

If \(S\) is a matrix in scalar-triangular form with eigenvalue \(\lambda = \lambda_0\) repeated down the diagonal, then \(\lambda I - S\) will have form like

\begin{align*} N_2 \amp = \begin{bmatrix} 0 \amp \ast \\ 0 \amp 0 \end{bmatrix} \text{,} \amp N_3 \amp = \begin{bmatrix} 0 \amp \ast \amp \ast \\ 0 \amp 0 \amp \ast \\ 0 \amp 0 \amp 0 \end{bmatrix} \text{,} \amp N_4 \amp = \begin{bmatrix} 0 \amp \ast \amp \ast \amp \ast \\ 0 \amp 0 \amp \ast \amp \ast \\ 0 \amp 0 \amp 0 \amp \ast \\ 0 \amp 0 \amp 0 \amp 0 \end{bmatrix} \text{,} \end{align*}

or similarly for larger sizes.

(a)

Compute power \(N_2^2\text{.}\)

Compute powers \(N_3^2\text{,}\) \(N_3^3\text{.}\)

Compute powers \(N_4^2\text{,}\) \(N_4^3\text{,}\) \(N_4^4\text{.}\)

What is the pattern?

(b)

What are the eigenvalues of each of \(N_2\text{,}\) \(N_3\text{,}\) \(N_4\text{?}\) What are the algebraic multiplicities of these eigenvalues?

What is the characteristic polynomial of each of \(N_2\text{,}\) \(N_3\text{,}\) \(N_4\text{?}\)

What is the pattern?

Discovery 31.4.

Repeat the tasks of Discovery 31.2 using the matrix

\begin{equation*} T = \left[\begin{array}{rrrrr} 5 \amp \ast \amp \ast \\ 0 \amp 5 \amp \ast \\ 0 \amp 0 \amp 5 \\ \amp \amp \amp -1 \amp \ast \\ \amp \amp \amp 0 \amp -1 \end{array}\right] \end{equation*}

in place of \(S\text{,}\) treating the \(\ast\) entries as “don't care” values.

When you compute \(c_T(T)\text{,}\) you may wish to use the patterns you discovered in Task 31.1.c and Discovery 31.2 to speed things up.

Discovery 31.5.

Based on Discovery 31.4, make a conjecture about the relationship between a matrix \(A\) and its characteristic polynomial \(c_A(\lambda)\text{.}\) In doing this, you should consider the following:

  • every matrix is similar to a triangular block matrix (at least over \(\C\text{,}\) anyway), and
  • the pattern of Task 31.1.b.
Discovery 31.6.

Again, for this discovery activity you should keep in mind that if we work over \(\C\text{,}\) then every matrix is similar to one in triangular block form (Theorem 30.5.1).

(b)

The characteristic polynomial of a matrix always factors (over \(\C\)) as

\begin{gather} c(\lambda) = (\lambda - \lambda_1)^{m_1} (\lambda - \lambda_2)^{m_2} \dotsm (\lambda - \lambda_\ell)^{m_\ell}\text{,}\label{equation-cayley-hamilton-discovery-general-char-poly}\tag{\(\star\)} \end{gather}

where the \(\lambda_j\) are the eigenvalues of the matrix and the \(m_j\) are the corresponding algebraic multiplicities.

If you were to expand the characteristic polynomial out completely again, what would the constant term be? Can you make a connection between this pattern and that of Task a?

(c)

Recall that the trace of a matrix is the sum of the diagonal entries. It turns out that similar matrices also have the same trace. Similar to Task a, make a conjecture about the relationship between the eigenvalues and the trace of a matrix.

(d)

Similar to Task b, if you were to expand the characteristic polynomial in (\(\star\)) out completely again, what would the coefficient on the \(\lambda^{n-1}\) term be? Can you make a connection between this pattern and that of Task c?