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Exercises

Exercise 6.63:

$A=\begin{bmatrix}5&-3&-6\\ 4&-2&-6\\ 2&-1&-3\end{bmatrix}$
$B=\begin{bmatrix}4&0&-1\\ -4&2&2\\ 2&0&1\end{bmatrix}$
$C=\begin{bmatrix}2&3&-1\\ -1&-1&1\\ 1&1&-1\end{bmatrix}$
$D=\begin{bmatrix}2&2&-1\\ -1&-1&1\\ -1&-2&2\end{bmatrix}$
$E=\begin{bmatrix}7&1&2&2\\ 1&4&-1&-1\\ -2&1&5&-1\\ 1&1&2&8\end{bmatrix}$ (You may assume that $c_{E}(x)=(x-6)^{4}$ )

You may assume that all eigenvalues of these matrices are integers (they were constructed this way for ease of computation, but this will not be true in general). For each of these matrices :

i.

Find the characteristic polynomial and minimal polynomial.
ii.

For each eigenvalue, find a basis for every generalized eigenspace $V_{\lambda}^{(i)}$ .
iii.

Find the JNF.
iv.

Find a Jordan basis.

Exercise 6.64:

Assume a matrix $A$ has the following characteristic polynomial. Find all possible JNF’s for $A$ (up to reordering of the Jordan blocks).

i.

$(x-1)^{2}(x+2)^{2}$ ,
ii.

$(x-1)^{3}(x+2)$ .

Exercise 6.65:

Let $A=\begin{bmatrix}a&b\\ c&d\end{bmatrix}\in\operatorname{M}_{{2}}({F})$ be a $2\times 2$ matrix over a field.

i.

Compute $c_{A}(\lambda)$ the characteristic polynomial.
ii.

Prove the Cayley-Hamilton theorem for all $2\times 2$ matrices.

Exercise 6.66:

Consider the function $T:\mathcal{P}_{2}({\mathbb{C}})\to\mathcal{P}_{2}({\mathbb{C}})$ defined by

T(f)(x)=f(x+1).

For example, $T(x^{2}+7)=(x+1)^{2}+7=x^{2}+2x+8$ .

•

Choose a basis $\mathcal{B}$ of $\mathcal{P}_{2}({\mathbb{C}})$ , and find the matrix ${}_{\mathcal{B}}[T]_{\mathcal{B}}$ .
•

Find the JNF of that matrix
•

Find a Jordan basis for $T$ ; this should form a basis of $\mathcal{P}_{2}({\mathbb{C}})$ .

Exercise 6.67:

Assume $A,B\in\operatorname{M}_{{n}}({{\mathbb{C}}})$ are similar matrices, and let $\lambda\in{\mathbb{C}}$ be an eigenvalue (by Exercise 4.58, $A$ and $B$ have the same eigenvalues). Prove that $\dim V_{\lambda}^{(i)}$ is the same for $A$ as it is for $B$ .

Exercise 6.68:

Let $D:\mathcal{P}_{3}({\mathbb{C}})\to\mathcal{P}_{3}({\mathbb{C}})$ be the differentiation transformation.

i.

Choose a basis $\mathcal{B}$ of $\mathcal{P}_{3}({\mathbb{C}})$ , and write down the matrix ${}_{\mathcal{B}}[D]_{\mathcal{B}}$ .
ii.

What is the minimal polynomial of $D$ ?
iii.

Find the JNF of $D$ .

Exercise 6.69:

A student is asked to prove Theorem 6.8(i). He writes the following:

[Student box]

Take two monic polynomials $p_{1},p_{2}\in\mathcal{P}(F)$ of minimal degree such that $p_{1}(A)=p_{2}(A)=\vec{0}$ , and assume $p_{1}\neq p_{2}$ . Since $p_{1}$ and $p_{2}$ are both of the same degree $r$ , and are monic, the polynomial $p_{1}-p_{2}$ is monic of degree $r-1$ . But notice that

(p_{1}-p_{2})(A)=p_{1}(A)-p_{2}(A)=\vec{0}-\vec{0}=\vec{0}.

So $p_{1}-p_{2}$ contradicts the minimality of $r$ . Therefore, our assumption $p_{1}\neq p_{2}$ must have been false. This proves $p_{1}=p_{2}$ , and in other words, the polynomial in the theorem is unique.

[End of Student box]

This student has made a logical mistake. What is it, and how could it be fixed?

Exercise 6.70:

A student is asked to prove the Cayley-Hamilton theorem (which is not an easy thing to do, and is omitted from this module). He writes the following:

[Student box]

Substitute $\lambda$ with $A$ in the characteristic polynomial. Then

c_{A}(A)=\det(A-A\cdot I_{n})=\det(\vec{0})=0.

Therefore, $c_{A}(A)=0$ for any square matrix $A$ , as required.

[End of Student box]

Identify the student’s mistake. You do not have to give a correct proof.

Exercise 6.71:

Prove that a matrix is diagonalizable if and only if

m_{A}(x)=(x-a_{1})(x-a_{2})\cdots(x-a_{r}),

where $a_{i}\neq a_{j}$ for $i\neq j$ .

Exercise 6.72 (Bonus):

Let $A\in\operatorname{M}_{{n}}({{\mathbb{C}}})$ , and consider the set of all matrices which are similar to $A$ . This is called the orbit of $A$ under the conjugation action. [ Aside: The words “orbit”, “conjugation”, and “action” will all be defined in MATH321.]

How many different orbits are there, among nilpotent $5\times 5$ complex matrices? Recall the definition of “nilpotent” from Exercise 4.64.

Learning objectives for Chapter 6:

Pass Level: You should be able to…

•

Verify the Cayley-Hamilton theorem for specific matrices (e.g. Exercise 6.4).
•

For a given (mostly factored) polynomial, produce a list of all possible monic factors which share the same roots (e.g. Exercise 6.13).
•

Given a matrix and its factored characteristic polynomial, find its minimal polynomial (e.g. Exercise 6.15).
•

Be able to express any generalized eigenspace of a matrix as the kernel of another matrix (e.g. Definition 6.17).
•

Given the generalized eigenspaces of a matrix, deduce its JNF using Theorem 6.48 (e.g. Exercise 6.50).
•

Correctly answer, with full justification, at least 50% of the true / false questions relevant to this Chapter.

First class level: You should be able to…

•

Write a complete solution, without referring to any notes, to at least 80% of the exercises in this Chapter, and in particular the proof questions.
•

Correctly answer, with full justification, all of the true / false questions relevant to this Chapter.