## Notation for bilinear form $y^t M z$, where $M$ is a matrix and $y,z$ are vectors.

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I'm working on a problem where I need to consider a bilinear form of the form $y^t M z$ where $M$ is an $n$-by-$n$ real symmetric matrix and `$y,z \in \mathbb{R}^n$` are vectors. I also need to consider restricted forms of such a product, of the form $$\sum_{i,j=1}^n y_i m_{ij} z_j \mathbf{I}_{(y_i,z_j) \in E},$$ where $E$ is some subset of $\mathbb{R}^2$. We recover $y^t M z$ by taking $E=\mathbb{R}^2$. I want a common notation for $y^t M z$ and for these restricted sums, so I have been writing $y^t M z = \langle y,z\rangle_M$, and writing $\langle y,z\rangle_{M,E}$ for the restricted sum above.

I see nothing wrong with the notation I'm using. However, if there's a standard notation for such things that I am unaware of, I would like to know about it. Is there? If you know of another notation, can you give me a reference?

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I think it would be better (more standard) to call these bilinear forms rather than inner products. Also, did you really mean $R^2$? And what is the meaning of the bold-face I? I think that the pointed bracket notation may be misleading since it might make one expect it to be symmetric---or do you mean to restrict the matrix $M$ to be symmetric? – Dick Palais Nov 19 2010 at 17:52
Thanks for the suggestions, I will modify my question accordingly! Yes in my situation the matrix is in fact symmetric, I will mention this.  – Louigi Addario-Berry Nov 19 2010 at 18:03
Also: the boldface I is meant to be the indicator function of the set E. – Louigi Addario-Berry Nov 19 2010 at 18:05
Note that a real inner product is usually defined to be non-degenerate, i.e. the matrix $M$ has to be non-singular. If that's not always the case in your situation, then the notation $\langle,\rangle_M$ is misleading. And certainly, the notation $\langle,\rangle_{M,E}$ seems misleading to me since these guys are degenerate. For the former, I would prefer "bilinear form $B(y,z)$", but for the latter, even that would be misleading, since it is not bi-linear. – Alex Bartel Nov 20 2010 at 5:01
Thanks Alex. The point of my question is to have a consistent notation for both the former and the latter, do you have a thought about what one might use in this case?  – Louigi Addario-Berry Nov 20 2010 at 10:44
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