Uniqueness of diagonal representation of a quadratic form?

Dear Sir,

For question 1 of the homework sheet, when we diagonalise, are the diagonal forms always unique?

Reply:

The diagonal form is not unique. For example, we can take

A=
3 0
0 4

and change to the equivalent form P^tAP where

P=
2 0
0 3

to get

A’=
12 0
0 36

That is to say, when we diagonalize a quadratic form, the diagonal terms are not *intrinsic* to the form. They are basis- dependent. This is in contrast to the case of a linear transformation, where a change of basis corresponds to the change A’=P^{-1}AP and the diagonal form ends up with the *eigenvalues* of A down the diagonal. Having written that, note that the exercise asks not for any diagonal form, but the real and complex *canonical* form. These are unique. Recall that the complex canonical form depends only on the rank of the matrix. For the real canonical form, it will turn out that the ones, minus ones, and zeros on the diagonal correspond exactly to the positive, negative, and zero eigenvalues. Of course you don’t need to know this to do the exercise. You need only use the double operations.

MK

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