## 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

and if your vector space

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

?

Another example, if

and vector space

then

or

2) Canonical form:

In lecture notes it says, a symmetric bilinear form is positive definite iff its canonical form (over R) is . But in solutions to sheet 7 it says canonical form is . So is it trivial that 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 has been chosen the only requirment of is that

is a basis of . In your first question is the whole space, so you need only choose so that you end up with a basis for the whole space. So your choice would work, but so would , , , and so on. Any vector that’s independent of the first two will work. Similarly, in the second question, can consist of any vector independent of the two in .

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

But here, is the dimension of the vector space. So if the original form was written in terms of some variables, say , then the number of variables is the same as the in . Check the dimension of the vector space in that question on sheet 7.

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