Real and complex canonical forms

[alvielwj]

A question about how find the canonical forms over R and C.
An example, given a quadratic form,q(x,y,z)=x^2 + 2xy + 4yz + z^2
find the canonical forms over R and C.

First step,i get the matrix 1 2^0.5 0
2^0.5 0 2
0 2 1
then by doing the double operation
i get the identiy matrix.
the canonical form over R is diag(I_r, -I_s, O_t)
and the canonical form over C is diag(I_r,0_t)
is the canonical form unique?
what are the final anwsers?

I know that any matrix can be changed to Identity matrix or a matrix with a 0 row.
does it mean most matries have similar canonical form?

 

[fresh_42]

Canonical is no property which is defined in any way. It literally means: "How it is usually done (written in the canon)". So whenever there is a usual way to the goal, then it is said canonical. In this case a canonical form, better standard form, is the one which has no mixed terms $x_ix_j$ and only squares $x_i^2$.

See http://www.maths.qmul.ac.uk/~twm/MTH6140/la26.pdf