Matrices and Forms I

Yesterday, we defined a Hermitian matrix to be the matrix-theoretic analogue of a self-adjoint transformation. So why should we separate out the two concepts? Well, it turns out that there are more things we can do with a matrix than represent a linear transformation. In fact, we can use matrices to represent forms, as follows.

Let’s start with either a bilinear or a sesquilinear form $B\left(\underline{\hphantom{X}},\underline{\hphantom{X}}\right)$ on the vector space $V$ . Let’s also pick an arbitrary basis $\left\{e_i\right\}$ of $V$ . I want to emphasize that this basis is arbitrary, since recently we’ve been accustomed to automatically picking orthonormal bases. But notice that I’m not assuming that our form is even an inner product to begin with.

Now we can define a matrix $b_{ij}=B(e_i,e_j)$ . This completely specifies the form, by either bilinearity or sesquilinearity. And properties of such forms are reflected in their matrices.

For example, suppose that $H$ is a conjugate-symmetric sesquilinear form. That is, $H(v,w)=\overline{H(w,v)}$ . Then we look at the matrix and find

$\displaystyle\begin{aligned}h_{ij}&=H\left(e_i,e_j\right)\\&=\overline{H\left(e_j,e_i\right)}\\&=\overline{h_{ji}}\end{aligned}$

so $H$ is a Hermitian matrix!

Now the secret here is that the matrix of a form secretly is the matrix of a linear transformation. It’s the transformation that takes us from $V$ to $V^*$ by acting on one slot of the form, and written in terms of the basis $e_i$ and its dual. Let me be a little more explicit.

When we feed a basis vector into our form $B$ , we get a linear functional $B(e_i,\underline{\hphantom{X}})$ . We want to write that out in terms of the dual basis $\left\{\epsilon^j\right\}$ as a linear combination

$\displaystyle B(e_i,\underline{\hphantom{X}})=b_{ik}\epsilon^k$

So how do we read off these coefficients? Stick another basis vector into the form!

$\displaystyle\begin{aligned}B(e_i,e_j)&=b_{ik}\epsilon^k(e_j)\\&=b_{ik}\delta^k_j\\&=b_{ij}\end{aligned}$

which is just the same matrix as we found before.

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June 24, 2009 - Posted by John Armstrong | Algebra, Linear Algebra

12 Comments »

At last: something for Physics.

Comment by Jonathan Vos Post | June 24, 2009 | Reply
DO you mean “Hamiltonian matrix” or “Hermitian matrix”?

Comment by Blaise Pascal | June 24, 2009 | Reply
Sorry, Blaise I have no idea where that came from.. fixing.

Comment by John Armstrong | June 24, 2009 | Reply
Have you treated (or do you plan to treat) forms (p-forms, differential forms)? Your style of brief explanations with lots of examples would be very helpful in providing a basic understanding of this often-overlooked tool.

Comment by Charlie C | June 25, 2009 | Reply
Charlie, I haven’t even done calculus in more than one variable yet.

But! That’s because I want some linear algebra at hand to use where it comes up in multivariable calculus, including just the sort of subjects you mention.

Comment by John Armstrong | June 25, 2009 | Reply
Shouldn’t $B(e_i,_)=\sum_k b_{ik}\epsilon^k$ or is that the standard way to write it (some kind of Einstein’s notation)?

Comment by watchmath | June 25, 2009 | Reply
Yes, and I just noticed that I forgot to hit “publish” on today’s post, which mentions something closer to that.

Comment by John Armstrong | June 25, 2009 | Reply
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