The Unapologetic Mathematician

Mathematics for the interested outsider

Nondegenerate Forms II

Okay, we know what a nondegenerate form is, but what does this mean for the transformation that represents the form?

Remember that the form represented by the transformation B is nondegenerate if for every nonzero ket vector \lvert v\rangle there is some bra vector \langle w\rvert so that \langle w\rvert B\lvert v\rangle\neq0. But before we go looking for such a bra vector, the transformation B has turned the ket vector \lvert v\rangle into a new ket vector B\lvert v\rangle=\lvert B(v)\rangle. If we find that B(v)=0, then there can be no suitable vector w with which to pair it. So, at the very least, we must have B(v)\neq0 for every v\neq0. That is, the kernel of B is trivial. Since B is a transformation from the vector space V to itself, the rank-nullity theorem tells us that the image of B is all of V. That is, B must be an invertible transformation.

On the other hand, if B is invertible, then every nonzero ket vector \lvert v\rangle becomes another nonzero ket vector \lvert w\rangle=B\lvert v\rangle. Then we find that

\displaystyle\langle w\rvert B\lvert v\rangle=\langle w\vert w\rangle>0

where this last inequality holds because the bra-ket pairing is an inner product, and is thus positive-definite. Indeed, a positive-definite (or negative-definite) form must be nondegenerate. Thus it is sufficient for B to be invertible.

Incidentally, this approach gives us a good way of constructing a lot of positive-definite transformations. Given an invertible transformation B, we expand

\displaystyle\langle w\vert w\rangle=\langle v\rvert B^*B\lvert v\rangle

Since the form defined by the bra-ket pairing is invertible, so is the form defined by B^*B. And this is a sensible concept, since B^*B is self-adjoint. Indeed, we take its adjoint to find

\displaystyle\left(B^*B\right)^*=B^*\left(B^*\right)^*=B^*B

This extends our analogy with the complex numbers. An invertible transformation composed with its adjoint is a self-adjoint, positive-definite transformation, just as a nonzero complex number multiplied by its conjugate is a real, positive number.

July 17, 2009 Posted by | Algebra, Linear Algebra | 2 Comments

   

Follow

Get every new post delivered to your Inbox.

Join 393 other followers