Now that we’ve got bilinear forms, let’s focus in on when the base field is . We’ll also add the requirement that our bilinear forms be symmetric. As we saw, a bilinear form corresponds to a linear transformation . Since is symmetric, the matrix of must itself be symmetric with respect to any basis. So let’s try to put it into a canonical form!
We know that we can put into the almost upper-triangular form
but now all the blocks above the diagonal must be zero, since they have to equal the blocks below the diagonal. On the diagonal, the blocks are fine, but the blocks must themselves be symmetric. That is, they must look like
which gives a characteristic polynomial of for the block. But recall that we could only use this block if there were no eigenvalues. And, indeed, we can check
The discriminant is positive, and so this block will break down into two blocks. Thus any symmetric real matrix can be diagonalized, which means that any symmetric real bilinear form has a basis with respect to which its matrix is diagonal.
Let be such a basis. To be explicit, this means that , where the are real numbers and is the Kronecker delta — if its indices match, and if they don’t. But we still have some freedom. If I multiply by a scalar , we find . We can always find some so that , and so we can always pick our basis so that is , , or . We’ll call such a basis “orthonormal”.
The number of diagonal entries with each of these three values won’t depend on the orthonormal basis we choose. The form is nondegenerate if and only if there are no entries on the diagonal. If not, we can decompose as the direct sum of the subspace on which the form is nondegenerate, and the remainder on which the form is completely degenerate. That is, for all . We’ll only consider nondegenerate bilinear forms from here on out.
We write for the number of diagonal entries equal to , and for the number equal to . Then the pair is called the signature of the form. Clearly for nondegenerate forms, , the dimension of . We’ll have reason to consider some different signatures in the future, but for now we’ll be mostly concerned with the signature . In this case we call the form positive definite, since we can calculate
The form is called “positive”, since this result is always nonnegative, and “definite”, since this result can only be zero if is the zero vector.
This is what we’ll call an inner product on a real vector space — a nondegenerate, positive definite, symmetric bilinear form . Notice that choosing such a form picks out a certain class of bases as orthonormal. Conversely, if we choose any basis at all we can create a form by insisting that this basis be orthonormal. Just define and extend by bilinearity.