Double operations

Evening Mr Kim,

Admittedly through my own fault, i haven’t been to a few of the lectures where you’ve taught us how to find canonical forms of matrices. All i know is that it’s something to do with double operations. If you have time, would you be so kind as to give me a brief guide on how to do this (for both R and C), particularly with regard to this weeks homework (sheet 7).

Here we have the quadratic forms so i have found the matrix corresponding to it through

q(x y z) = (x y z)(A)(x y z)^t

which I hope is the correct first step. Now i have a matrix with which i have no idea what to do. i.e. for the first one i have {(1,1,0) (1,0,2) (0,2,1)}

Thanks in advance for any advice,

Sam Popplestone

Reply:

Your first step is definitely correct. Read one of the earlier questions for hints on how to find A. For the remainder, you should definitely read lectures 20, 21, and 22 of the notes and the remark on quadratic forms. In brief, you need to diagonalize using double operations that consist of column operations followed by the `corresponding’ row operations. For example, if you add column 1 to column 2, you should follow by adding row 1 to row 2.

[(1 ,-1), (-1,-2)] —–> [(1 , 0 ), (-1,-3)]—–> [(1 , 0), (0,-3)]

It’s very easy to get used to. Just the first few examples in lecture 21 should get you going. After you’ve found the diagonal form, arrange the diagonals from top-left to bottom-right in the order of positive terms, negative terms, then zeros. You see, you can switch the order of any two diagonals, say, the (i,i) entry and the (j,j) entry, by using the double operation that switches column i and column j, and then switches row i and row j. Once you’ve done that, you get the real canonical form by converting all the positive terms into 1’s and the negative terms into -1’s. For example, in the above, you don’t need a rearrangement and there are no zeros. The real canonical form is

1 0

0 -1

The complex canonical form is then obtained by changing the minus 1’s to plus 1’s. Thus,

1 0

0 1

in our example.

It is explained in lemma 4.4.104 why these last two steps lead to congruent matrices. Make sure you understand it.

MK

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