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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 .
Discovery 31.3 .
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
\(N_4^2\text{,}\) \(N_4^3\text{,}\) \(N_4^4\text{.}\)
(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
\(N_2\text{,}\) \(N_3\text{,}\) \(N_4\text{?}\)
Discovery 31.4 .
\begin{equation*}
T = \begin{abmatrix}{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{abmatrix}
\end{equation*}
in place of \(S\text{,}\) treating the \(\ast\) entries as “don’t care” values.
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:
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 ).
(a) Make a conjecture about the relationship between the eigenvalues and the determinant of a matrix.
(b) Make a conjecture about the relationship between the eigenvalues and the
trace of a matrix.
(c)
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{,}\tag{✶}
\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 ?
(d) Similar to
Task c , if you were to expand the characteristic polynomial in
(✶) 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 b ?
