Discovery guide 29.2 Discovery guide
If \(P\) is a square matrix, write \(\uvec{p}_1, \uvec{p}_2, \dotsc, \uvec{p}_n\) for the columns of \(P\text{,}\) so that \(P = \begin{bmatrix} \uvec{p}_1 \amp \uvec{p}_2 \amp \cdots \amp \uvec{p}_n \end{bmatrix} \text{.}\)
For this discovery guide, we will need to recall a few important things.
- An \(n \times n\) matrix is invertible if and only if its columns form a basis of \(\R^n\) (Theorem 21.5.5).
- Similarity relation \(\inv{P} A P = B\) holds if and only if each column of \(B\) consists of coefficients for expressing the corresponding transformed vector \(A \uvec{p}_j\) as a linear combination of the columns of \(P\) (Subsection 26.3.2).
- Vector \(\uvec{x}\) in \(\R^n\) is an eigenvector of \(A\) if \(A\uvec{x} = \lambda \uvec{x}\text{.}\) Equivalently, \(\uvec{x}\) is an eigenvector of \(A\) if \((\lambda I - A) \uvec{x} = \uvec{0}\text{.}\)
Discovery 29.1.
Suppose \(A\) is a \(3\times 3\) matrix with characteristic polynomial \(c_A(\lambda) = (\lambda - 5)^3 \text{.}\)
(a)
What are the eigenvalues of \(A\text{?}\) What are the algebraic multiplicities of its eigenvalues?
(b)
If \(A\) is similar to a triangular matrix \(T\) (upper or lower, whichever you like), how many entries of \(T\) can you be sure about? Write down an example matrix \(T\) to demonstrate your answer.
See Theorem 26.5.8 and Proposition 24.6.1.
(c)
Do you remember what a scalar matrix is? Why do you think your example matrix \(T\) from Task b is called scalar-triangular form?
Break the form matrix \(T\) apart into the sum of scalar part and a “purely” triangular part.
Discovery 29.2.
The matrix \(A\) below is similar to a matrix of the form \(B\) below:
(Treat the \(\ast\) entries in \(B\) as “don't care” values.)
Let \(P\) be a transition matrix that realizes the similarity \(\inv{P}AP = B\text{.}\) As usual, we would like to determine the conditions on the columns of \(P\) that create the similarity relationship between \(A\) and \(B\text{.}\)
(a)
(i)
Use the first column of \(B\) to express \(A\uvec{p}_1\) as a linear combination of \(\uvec{p}_1,\dotsc,\uvec{p}_5\text{.}\)
(ii)
Identify the pattern: The first column \(\uvec{p}_1\) must be .
(iii)
Now use your pattern and some row reducing to actually compute a possible \(\uvec{p}_1\) that could be used.
(b)
Remembering that the \(\ast\) entries in \(B\) are “don't care” values, could you use the same kind of vector for \(\uvec{p}_2\) as for \(\uvec{p}_1\text{,}\) and still get the proper form for the second column of \(B\text{?}\)
Looking back at how you computed \(\uvec{p}_1\) in Task a, is it even possible to obtain such a \(\uvec{p}_2\text{?}\)
(c)
Repeat Task b for \(\uvec{p}_3\text{,}\) but this time you need to consider the third column of \(B\text{.}\)
(d)
(i)
Use the third column of \(B\) to express \(A \uvec{p}_3\) as a linear combination of \(\uvec{p}_1,\dotsc,\uvec{p}_5\text{.}\)
(ii)
Rearrange your equality expressing \(A \uvec{p}_3\) as a linear combination into an expression
where \(\lambda\) is the shared single eigenvalue of \(A\) and \(B\text{.}\) (Did you realize you did something similar to compute \(\uvec{p}_1\text{?}\) See the reminder about eigenvalues and eigenvectors in the introduction to this worksheet.)
(iii)
From the way you obtained \(\uvec{p}_1\) and \(\uvec{p}_2\) in Task a and Task b, what space do those first two vectors span?
And so it must be that the vector \(\uvec{w} = (\lambda I - A) \uvec{p}_3\) is .
(iv)
From the property of \(\uvec{w}\) you identified, this vector must satisfy the homogeneous matrix equation . (Again, see the introduction of this worksheet.)
And since \(\uvec{w} = (\lambda I - A) \uvec{p}_3\text{,}\) that means \(\uvec{p}_3\) must satisfy the homogeneous matrix equation .
Now solve for \(\uvec{p}_3\text{.}\)
(e)
Repeat Task d.i and Task d.ii for \(\uvec{p}_4\) (considering the fourth column of \(B\) now).
We found \((\lambda I - A) \uvec{p}_3\) needed to be in the span of \(\uvec{p}_1\) and \(\uvec{p}_2\text{.}\) What span does \((\lambda I - A) \uvec{p}_4\) need to be in? Will it be okay if it is forced to also be in the span of just \(\uvec{p}_1\) and \(\uvec{p}_2\) instead? (Remember that the \(\ast\) entries of \(B\) are “don't care” values.)
If so, the computation you already performed in Task d.ii might provide you with another vector to use as \(\uvec{p}_4\text{.}\)
(f)
Can the same reasoning used in Task e be used to obtain a suitable vector for \(\uvec{p}_5\text{?}\)
(g)
If you've made it this far, repeat the kind of reasoning that we used to determine \(\uvec{p}_3\) in Task d to figure out how to solve for a suitable \(\uvec{p}_5\text{.}\)
Discovery 29.3.
Summarize the pattern of how you determined the columns of \(P\) in Discovery 29.2.
The collection of vectors that are in the null space of \((\lambda I - A)^k\) for at least one positive exponent \(k\) is called the generalized eigenspace of \(A\) for the eigenvalue \(\lambda\text{,}\) and is denoted \(G_{\lambda}(A)\text{.}\)
Discovery 29.4.
Suppose \(A\) is an \(n \times n\) matrix and \(\lambda\) is an eigenvalue of \(A\text{.}\)
Use the Subspace Test to verify that \(G_{\lambda}(A)\) is a subspace of \(\R^n\text{.}\)
Careful: When you check closure under addition, you cannot assume that both arbitrary vectors from \(G_{\lambda}(A)\) are in the null space of the same power of \((\lambda I - A)\text{.}\)
Discovery 29.5.
It is quite easy to switch between upper and lower triangular forms. Consider the transition matrix
(a)
Verify that \(\inv{J} = J\text{.}\)
(b)
Use the pattern of similarity described in the introduction of this discovery guide to show that if \(U\) is an upper triangular matrix of the same size as \(J\text{,}\) then \(\inv{J}UJ\) is lower triangular.
(c)
Show that if \(L\) is a lower triangular matrix of the same size as \(J\text{,}\) then \(\inv{J}LJ\) is upper triangular.