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 belongs to , then belongs to for greater than 1. I have noticed that sometimes there is a choice as to which basis vector in to replace. Does it matter? Is there a convention? Similarly, when extending our basis

for

to be a basis for , 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, and Say the characteristic polynomial is

and the minimal polynomial is

with, obviously, less than or equal to , less than or equal to . I don’t understand how we know without proof that . 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

will not have a Jordan canonical forms over . Over , 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 with one eigenvalue , the requirement is that

for . 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 , we have

and hence,

Therefore, is stabilized (that is, taken to itself) by . Similarly, is stabilized by . By considering the shape of the Jordan canonical form for

(the one eigenvalue case) we see that

for . Similarly,

for . But we have

by the primary decomposition theorem. So

or

Therefore,

and

.

 

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,

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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?

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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 ,

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 and not suffice?

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

Thanks

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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 and 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 by substitution would involve.

 

Question on discriminants

Hi Professor Kim hope you’re doing well. I just had a quick question to do with the notes on lecture 12 on the pdf file, in lecture 12 theres a ‘proof for the theorem’, I was just wandering is this an alternative proof for theorem 106? Or is it part of the same proof but just looking at the Norm of the derivative of the minimal polynomial?

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Reply:

I’m sorry. That portion of the notes is a bit disorganized. You are right that it is just an alternative proof. Thanks for pointing this out.

 

Canonical forms and equivalence

Hi Sir,

just to make sure if I’m correct, is it true that if 2 symmetric matrices
have the same real and/or complex canonical forms then they are equivalent?
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Reply:

To be precise, two symmetric matrices are equivalent over if and only if they have the same real canonical form. They are equivalent over if and only if they have the same complex canonical form.

 

Question on isometries

Dear Professor,

Firstly thanks for taking the time out to reply to our messages, we really appreciate the help and guidance from yourself. I wanted to ask the following short question about the 2201 notes:

On page 56 Thm 5.4.150: after showing that and , can’t we simply show by saying:

and proceeding exactly as the last line of the proof at the bottom?

Thanks in advance.

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Reply:

Not quite. Although with a bit more argument it might be OK. In `the last line of the proof at the bottom,’ note that an unwritten step in the argument says

for *all* , and then one sets . This wouldn’t be admissible if we just had . Try it out to see clearly for yourself what I mean.

 

Internet

One other important point:

I’ve been having a rather exasperating 10 days trying to get BT to activate my broadband service in a new house, sadly unsuccessfully. So I may not be able to get to email questions until Monday afternoon. The people who intended to send them could also avail themselves of the special office hour.

 

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…

 

Proofs

I thought I would post a short email exchange, just in case other people are confused by this issue as well.

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Email:

hi, this is where the confusion arose as you mention some proofs specifically but at the bottom where it says “Needless to say, it is assumed that you will have a full
understanding of the material surrounding the results listed above, especially the definitions and the examples.” does this include the proofs of the theorems?

>
> For the answer to this, look at the course summary on the course webpage.
>
> Best,
>
> MK
>

>> do we need to know the proofs that are in the online notes but not in the
>> class notes?

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The answer to the last question is: Yes, of course you need to know the proofs. The paragraph quoted was put in to emphasize that you need to know *more* than the proofs. The obvious intention was to discourage people from pure memorization, which has been a topic of discussion since I taught the course last year. Let me once again refer people to the the posts on the blog surrounding last year’s exam. The exchange reproduced above seems to illustrate once again a tendency to indulge in a `minimal reading’ of requirements. Be careful. to spell it out once more, what is meant is:

`You need to know the proofs and understand them fully. This means you need *also* to understand the material surrounding the theorems and their proofs.’