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 on the vector space . Let’s also pick an arbitrary basis of . 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 . This completely specifies the form, by either bilinearity or sesquilinearity. And properties of such forms are reflected in their matrices.
For example, suppose that is a conjugate-symmetric sesquilinear form. That is, . Then we look at the matrix and find
so 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 to by acting on one slot of the form, and written in terms of the basis and its dual. Let me be a little more explicit.
When we feed a basis vector into our form , we get a linear functional . We want to write that out in terms of the dual basis as a linear combination
So how do we read off these coefficients? Stick another basis vector into the form!
which is just the same matrix as we found before.