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.

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July 8, 2009 - Posted by | Algebra, Linear Algebra

4 Comments »

  1. Should the line 3 say, “Now, any linear transformation B:V–>V”?

    Comment by Adam Glesser | July 9, 2009 | Reply

  2. Yes, fixed

    Comment by John Armstrong | July 10, 2009 | Reply

  3. [...] Antisymmetric, and Hermitian Forms The simplest structure we can look for in our bilinear forms is that they be symmetric, antisymmetric, or (if we’re working over the complex numbers) [...]

    Pingback by Symmetric, Antisymmetric, and Hermitian Forms « The Unapologetic Mathematician | July 10, 2009 | Reply

  4. [...] work with, we need to pick out a bilinear form (or sesquilinear, over . So this means we need a transformation to stick between bras and [...]

    Pingback by Positive-Definite Transformations « The Unapologetic Mathematician | July 13, 2009 | Reply


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