More on canonical forms

Dear Professor Kim,

I encountered some problems when practising on the past papers.

1. In the proof of the Primary Decomposition Theorem, since the minimal polynomial m(T)=0, then Ker(m(T))=V, why is that?

2. I am still confused with how to use rank and null to decide a Jordan basis.

3. In the final process of finding the Real/Complex Jordan Canonical Form, if I have to multiply matrices E&(E transpose) with complex elements to get the Real JCF, is it right or I made a mistake (problem is I’ve checked several times that I didn’t make any mistake in the previous calculation)?

4. Relating to the previous question, is it possible to have real JCF and complex JCF the same?

Finally, since I’m in China right now I can’t access the web blog. Would you please copy the answers to my email? Thanks a lot!!

Reply:

1. As you say, if m(x) is the minimal polynomial, then m(T) is the *zero* endomorphism. What does it mean for the endomorphism to be zero?

2. The whole topic of computing Jordan bases is important, but hard to describe on the blog. It’s important to ask very specific questions. However, I can guess what difficulty you’re trying to express. For an endomorphism T and each eigenvalue a, we compute the Jordan basis corresponding to that eigenvalue. In the process, very relevant things to analyze are the kernels of

(T-aI)^k

for various k. For example, for k=1, the kernel is exactly the eigenspace corresponding to a. (Make sure you understand why this is so. I still find many people confused on this elementary point.) For k>1, we call these kernels the generalized eigenspaces. Also very important is that these kernels are nested:

Ker(T-aI) \subset Ker(T-aI)^2 \subset Ker(T-aI)^3 \subset \cdots

This also, make sure you understand why. The final important general relation is that

(T-aI)(Ker(T-aI)^{k+1})\subset (T-aI)^{k}

Anyways, in terms of the Jordan basis that you’ll see arranged in an array either in the notes or in my notes for 12-11-07, the bottom row is a basis for the kernel of (T-aI). The bottom *two* rows is a basis for the kernel of (T-aI)^2, and so on. Thus, the nullities of these powers of (T-aI) encode information about the shape of that array, from which one can extract the precise Jordan canonical form. Remind yourself precisely what the heights of the columns mean. These heights are exactly what you need for the precise Jordan canonical form.

Now to your question: Sometimes, you’ll see yourself given the *rank* of (T-aI)^k rather than its nullity. But don’t you know how to recover one from the other?

3. As a matter of terminology, the real or complex canonical forms of a symmetric bilinear form (or a quadratic form) are *not* called the *Jordan* canonical forms. If your confusion of the two reflects a conceptual confusion, you should clear it up right away. One very important distinction is that *similarity* of matrices is different from *congruence* of matrices. Look this up.

Anyways, *no*, you should not use complex matrices when finding the real canonical form even though it might luckily give you the right answer. The reason is that the real canonical form refers to a basis for R^n, and if you use complex transformations, the basis you end up with may be a basis for C^n.

4. The answer is yes. And I’m going to ask you back the question `*when* are they the same?’ Make sure you know the answer and the reason for it.

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