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?