Real canonical form


I am confused on one thing.

For the first quadratic form in question 1, I end up with the matrix


On your blog you have written “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.”

Does that mean the real canonical form is:

{(1,0,0),(0,1,0),(0,0,-1)} ????????


Yes. The real canonical form, by definition, is of the form

diag(I_r, -I_s, O_t)

where I_r denotes the identity r-by-r matrix, -I_s, minus the identity s-by-s matrix, and O_t, the 0 t-by-t matrix. For uniqueness, the order in which they are put along the diagonal must be fixed, and the convention is as described. So if the signs along the diagonal are not in the right order, they first have to be interchanged. As described in class, this can be achieved using a double operation. And then, of course, a diagonal term `a’ can be changed to `c^2a’ for any c (again using double operations). Over R, this means that any positive number can be changed to 1, and any negative number to -1.



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