The Unapologetic Mathematician

Mathematics for the interested outsider

Matrices and Bilinear Forms on Inner Product Spaces in Dirac Notation

Now, armed with Dirac notation, we can come back and reconsider matrices and forms. For our background, we’ve got an inner product space. That is, a vector space V, equipped with a choice of a particular inner product \langle\underline{\hphantom{X}},\underline{\hphantom{X}}\rangle.

Now, any linear transformation B:V\rightarrow V gives us a bilinear form. In our new notation we can write it as B(v,w)=\langle v\rvert B\lvert w\rangle. Given a basis \left\{\lvert i\rangle\right\}_{i=1}^n we can write down the matrix b_{ij}=\langle i\rvert B\lvert j\rangle. Then if we’re given vectors \langle v\rvert=v^i\langle i\rvert (notice how the Dirac notation can be rather context-dependent) and \lvert w\rangle=\lvert j\rangle w^j, we can put them together with B to find

\displaystyle\begin{aligned}\langle v\rvert B\lvert w\rangle&=v^i\langle i\rvert B\lvert j\rangle w^j\\&=v^ib_{ij}w^j\end{aligned}

So this is indeed the same old matrix of the form.

We can read a lot of information about the form off of its matrix. As we proceed we’ll illustrate these various properties of bilinear forms, using the Dirac notation to (hopefully) make the ideas clearer.

July 8, 2009 Posted by | Algebra, Linear Algebra | 4 Comments