Category Archives: 2201

Long email

I received a long last minute email. I’ll just copy it together with my answers without proper formatting or corrections because I’m also a bit tired.

MK

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Because of lack of time, I will be brief with the answers:

1. Everything is OK except for the statement about the JCF and
the linearly independent eigenvector. A basis for the eigenspace
can be taken as 1. There is just one Jordan block. This should tell you
what the JCF is. (Obviously can’#t be zero).

2. You are right.

3. Right again.

4. The definition as given is correct. For example, might have
f=m. There must have been a misprint in the mark scheme.

5. Yes, these are the same. More precisely, Ker(A)=Ker(-A) for
obvious reasons.

6. One way of thinking about C is as R^2 with some multiplication.
Anyways, the dimension is 2 for the reason you say as well. The dimension
of any field F as a vector space over itself is one. The dimension of
R as a Q-vector space is infinite.

7. See what you can do with L^{98}v.

8. Yes to both questions.

9. This is somewhat tricky. Look at example 125 on the 3704 lecture notes for the general idea.

10. You could, but it might be quicker to take a general symmetric
matrix and show that it can’t work.

11. This is a bit complicated to explain by email in a short amount of
time.

> Dear Prof Kim,
>
> Sorry for the late email. I have quite a few questions on 2201 course
> material. I will be very grateful if you could clear my queries.
>
>
> COURSE SUMMARY AND LECTURE NOTES:
> 1) Summary sheet on course mentions ‘vector space of polynomials with
> complex coefficients of degree at most 5’ Would this be {i, ix, i(x^2),
> i(x^3), i(x^4), i(x^5)} So on D(differentiation):VtoV we get min
> polynomial=ch polynomial=x^5 and so JCF is the zero matrix? And so D has
> only 1 linearly independent eigenvector (i.e. 0)?
>
> 2) Lecture notes page 43: In the proof for diagonalisation theorem, 3
> lines before the end of proof, ‘Hence by the inductive hypothesis, there
> is an orthonormal basis’ shouldn’t this be orthogonal basis?
>
> 3) Page 47 proof of sylvesters law of inertia: In the new basis ci =
> bi/sqrt(q(bi)) and not bi/sqrt(q(bi,bi))?
>
> Also further in the proof ‘q(u)=x1^2+…+xr^2>0’ shouldn’t this be >=0 as
> only then further on U intersection W = {0}
>
> 4) Lecture notes page 25: Definition of minimal polynomial second
> condition on f(T)=0 and f=/0 then deg(f)>=deg(m). Shouldnt deg(f) be
> strictly > deg(m)as suggested in the 2008 mark scheme.
>
> 5) Definition of t-th generalised eigenspace is V(t)(la)=ker((la.Id-T)^t)
> (page27) but in primary decomposition theorem (page28) we use this def as
> V(t)(la)=ker((T-la.Id)^t). Would this not change signs? Are they the same
> because for e.g. kernel implies T(v)=0 so T(-v)=0 too.
>
> 6) What is the dimension of C as an R-vector space? Is this 2 because
> basis is {1, i}?
> What is the basis of C as a C-vector space?
> What is the basis of Q as a Q-vector space?
> What is the basis of R as a Q-vector space?
>
> 7) Suppose there exists a vector v such
> that (L)^100(v)=0,(L)^99(v)=/0. Prove that there exists a
> vector w such that (L)^2(w)=0 and (L)(w)=/0.
> What would be the steps for this proof?
>
>
> HOMEWORK SHEETS:
> 8) Sheet 2 qu7 – Do we only do one long division to get ans?
> Sheet 2 qu9 – Not on syllabus?
>
> 9) How to do sheet 3 qu7?
>
> 10) Sheet 6 qu4 – Can we do this by finding all sym bilinear forms on
> field 2 (8 matrices) and showing none of their quadratic forms are xy?
>
> 11) Sheet 7 qu4 – How do I do part a? Part b is simple. For part c, do we
> explicitly show this for the 9 different possibilities. Most of these
> already in required form (i.e. q00, q01, q10, q11, etc..)
>
>
> Sorry for so many questions. I don’t live near university. Otherwise I
> would have come in to see you. If its easier for you to reply over a
> telephone conversation, please call me on 07971530295.
>
> Thank you very much for your time.
>
>

More Jordan bases and canonical forms

Dear Professor,

I have a couple of questions about the Jordan canonical form section of the course if you get a spare moment.
1. I understand that if are considering linear maps from a vector space over the complex numbers to another vector space over the complex numbers, then eigenvalues always exist. When we find a Jordan basis, we use these eigenvalues. Does this mean we are assuming that the field in question is the complex numbers? Surely if not, then these eigenvalues aren’t guaranteed to exist so we can’t always find a Jordan basis?

2. After finding a pre-Jordan basis, we have to replace some basis vectors so as to satisfy the condition that if b_i belongs to V_i(\lambda), then (L -\lambda)b_i belongs to V_{i-1}(\lambda) for i greater than 1. I have noticed that sometimes there is a choice as to which basis vector in V_{i-1}(\lambda) to replace. Does it matter? Is there a convention? Similarly, when extending our basis

B_1\cup B_2\cup ...\cup B_{i-1 }

for

V_{i-1}(\lambda)

to be a basis for V_i(\lambda), there may be a choice as to which basis vector we choose to add. Does it matter? Is there a convention? My questions above lead me to believe that there are actually infinitely many Jordan bases for any one linear map / matrix, but that their Jordan canonical form is unique up to permutation of the Jordan blocks. Am I correct in thinking this?

3. Finally, say we have a linear map with two eigenvalues, \alpha and \beta. Say the characteristic polynomial is

(X - \alpha)^a (X - \beta)^b

and the minimal polynomial is

(X - \alpha)^c (X - \beta)^d

with, obviously, c less than or equal to a, d less than or equal to b. I don’t understand how we know without proof that dim V_c(\alpha) = a. I have chosen a two eigenvalue case for simplicity – obviously I’m interested in all several eigenvalue cases.

Thanks very much!

Reply:

1. You are right that that eigenvalues need to be contained in the field of definition for a linear map to admit a Jordan canonical form. For example, the rotation matrix

\left( \begin{array}{cc} 0 & -1 \\ 1 & 0  \end{array} \right)

will not have a Jordan canonical forms over R. Over C, however, it is diagonalizable, and the diagonal form *is* the Jordan canonical form. (What is it?)

2. There is an error in the condition you write, which could be minor or serious. To be a Jordan basis for a linear map L with one eigenvalue \lambda, the requirement is that

(L-\lambda)B_i\subset B_{i-1}

for i>1. Look carefully to see and understand the difference from what you wrote. (Actually, from the overall understanding reflected in your message, I suspect your mistake was just a misprint. But I wrote the above for the general reader.)

As to your question, you are write there there are many choices involved. This is the point people often find confusing, not just in this topic, but in many basic mathematical problems: when there is not a unique solution. We just have to understand the material well enough to feel relaxed about the choices. There is no general convention I can think of regarding `good’ choices, other than obvious demands of economy like simple numbers and as many zero entries as possible.

All of your remaining observations are correct. I wouldn’t be too surprised if a study of the *space of all possible Jordan bases* would yield some insight on good choices, at least in some natural special situations.

3. The general discussion is an obvious generalization of the case with two eigenvalues, so let’s stick to your question as it stands.

For v \in V_c(\alpha), we have

(L-\alpha )v \in V_{c-1}(\alpha)\subset V_c(\alpha)

and hence,

Lv\in \langle v \rangle+V_c(\alpha)=V_c(\alpha).

Therefore, V_c(\alpha) is stabilized (that is, taken to itself) by L. Similarly, V_d(\beta) is stabilized by L. By considering the shape of the Jordan canonical form for

L|V_c(\alpha)

(the one eigenvalue case) we see that

ch_{L|V_c(\alpha)}(X)=(X-\alpha)^s

for s=dim V_c(\alpha). Similarly,

ch_{L|V_d(\beta)}(X)=(X-\beta)^t

for t=V_d(\beta). But we have

V=V_c(\alpha)\oplus V_d(\beta)

by the primary decomposition theorem. So

ch_L(X)=ch_{L|V_c(\alpha)}(X)ch_{L|V_d(\beta)}(X)

or

(X-\alpha)^c(X-\beta)^d=(X-\alpha)^s(X-\beta)^t.

Therefore,

c=s=dim V_c(\alpha)

and

d=t=dim V_d(\beta) .

Spectral theorem etc.

Dear Professor Kim,

I have looked at your course summary and I cannot find anything about the Spectral Theorem. The proof of this theorem was not covered in class, and in the online notes it only says “See the article ‘The Spectral Theorem’ on the course webpage”. I have however seen that a question on this proof in 2006 paper, so I was wondering if we need still need to know this proof? Also, do we need to know the extra material on the course webpage such as the supplementary notes and articles?

Best regards,

———————————————

Reply:

The course summary definitely mentions the spectral theorem: On the list at the end, it is theorem 5.5.159, whose proof, as you point out, is in the supplementary notes. I hope the summary and the course blog is making the relative importance of different portions clear enough. Essentially all the supplementary material should be useful.
But perhaps most relevant are:

– A practical summary of the course

-The spectral theorem

-Remark on quadratic forms

-The first two pages of `Supplementary note on self-adjoint maps’

and the various example sheets on Jordan canonical forms.

Positive definite inner products

Hi Sir,

I’ve got another question.

When determining whether an inner product (in the Real space) is positive definite, we check it’s Real Canonical Form. If it’s diagonals are all “1”s, then it is positive definite. What about determining positive definiteness in the Complex space? I realised(from Sheet 8, Qns 1e) that the same method wouldn’t work (find the complex canonical form and see if its diagonal entries are all “1”s). How can we determine if an inner product is positive definite in the complex space?

—————————————————–

Reply:

Your observation is correct. Note that in the complex space, the positive definiteness is being checked for a *Hermitian* form, not a symmetric bilinear form (what is the difference?). Hence, the whole discussion of complex canonical forms does not apply. The only answer I can give at the moment is that you should think about the definition carefully. Of course there are the problems from the coursework, but it may help already to ask yourself: Is the standard Hermitian form on C^n,

\langle v,w \rangle := v_1\bar{w}_1+v_2\bar{w}_2+\cdots v_n\bar{w}_n

positive definite? Why or why not?

Question on Bezout’s Lemma

For a question such as on Homework sheet 3 1ii.) and on some of the past papers, is it strictly necessary to use Bezout’s lemma? Would a simple substitution of f and g not suffice?

Also, would you get full marks for using a substitution instead of using Bezout’s Lemma.

Thanks

———————————————————-

Reply:

You would get full marks for any correct proof or answer clearly written.

Note that Bezout’s lemma is merely an `explicit Euclidean algorithm.’ As soon as f and g get at all large, you’ll find that the Euclidean algorithm is much more efficient than direct substitution. In some suitably general sense, the Euclidean algorithm is one of the most efficient algorithms around. It’s a bit of a miracle, in some sense.

However, your question is a good one. Check for yourself what the procedure of finding h, k by substitution would involve.

Special office hour

In view of the 2201 exam, I will be in the fifth floor common room on Monday, 11 May, from 4 to 5 PM, to answer last minute questions. This is of course for your psychological comfort. In practice, you don’t really want to be in the position of relying on last minute questions…

More general questions on 2201 exam

Hi

I was just wanted to ask you about the structure of the exam next week? (as I missed the revision session during Easter)

1) what will the layout be like? similar to 2008 paper?

2) how come the 2008 paper is different from previous years, the notation and the method of asking the questions seems different?

3) are there any key areas that will come up?

4) there are 5 chapters in the notes, will there be a question from each chapter? which chapter will have 2 questions from it?

Thanks for your time and I look forward to your reply

——————————————————

Reply:

I believe most of questions 1 and 3 have been answered in previous blog entries and the course summary. Regarding question 5, the questions will be reasonably well-distributed across the chapters as you might expect from any exam of this nature. Surely, I can’t tell you that some specific chapter will have two questions from it! I will say that about three of the questions may be distributed across roughly two of the chapters.

Regarding your question 2, you should look at the blog entries from around exam time last year to get some insight on this issue. The short answer is that if the 2008 exam looks very different from previous years, then you haven’t studied enough. This I mean quite seriously. At the level of scientific research, of course a conceptual breakthrough occurs when disparate phenomena are seen to be manifestations of the same principle. But even in a basic course, sufficient understanding is pretty well measured by the ability to see that rather large collections of problems are actually minor variations on the same theme. Certainly, don’t expect that the exam problems will be the same as the coursework or those of previous years with just the numbers changed!

Pre-Jordan basis

Dear Prof Kim,

I had a couple of questions on Algebra3.

1) Pre-Jordan basis:

If you have your pre-jordan basis as

B_1=\{(-1,1,0)^t\}, B_2=\{(-1,0,1)^t\}

and if your vector space

V_3=Ker(0 \ \mbox{matrix})

then how do you decide the pre jordan basis B_3? Is it always

(1,0,0)^t?

Another example, if

B_1=\{(-1,0,3)^t,(-2,3,0)^t\}

and vector space

V_2=Ker (0\  \mbox{matrix})

then

B_2=\{(1,0,0) ^t\}

or

(0,1,0)^t?

2) Canonical form:

In lecture notes it says, a symmetric bilinear form is positive definite iff its canonical form (over R) is I_n. But in solutions to sheet 7 it says canonical form is I_{n+1}. So is it trivial that n can be any natural number depending on size original matrix corresponding to the sym bilinear form?

Thank you for your help.

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Reply:

Neither the pre-Jordan basis nor the Jordan basis are unique. This perhaps creates some difficulty in related questions owing to the answer not being completely determined.

Once B_1, B_2, \ldots B_i has been chosen the only requirment of B_{i+1} is that

B_1\cup B_2\cup \cdots \cup B_i\cup B_{i+1}

is a basis of V_{i+1}. In your first question V_3 is the whole space, so you need only choose B_3 so that you end up with a basis for the whole space. So your choice would work, but so would (0,1,0)^t, (0,0,1)^t, (1,1,1)^t, and so on. Any vector that’s independent of the first two will work. Similarly, in the second question, B_2 can consist of any vector independent of the two in B_1.

Regarding symmetric bilinear forms, it’s correct that f is positive definite if and only if its real canonical form is

I_n.

But here, n is the dimension of the vector space. So if the original form was written in terms of some variables, say f(x_1,x_2, \ldots, x_n), then the number of variables n is the same as the n in I_n. Check the dimension of the vector space in that question on sheet 7.

Course updates

The practical summaries for the courses 2201 and 3704 from the Fall term of 2008 have now been updated on the respective course webpages. In particular, a list of important results have been included.

The course notes for 2201 have now a few more lines on the diagonalization of quadratic forms.