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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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  • $\begingroup$ 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? $\endgroup$
    – Dick Palais
    Commented Nov 19, 2010 at 17:52
  • $\begingroup$ Thanks for the suggestions, I will modify my question accordingly! Yes in my situation the matrix is in fact symmetric, I will mention this. $\endgroup$
    – Louigi Addario-Berry
    Commented Nov 19, 2010 at 18:03
  • $\begingroup$ Also: the boldface I is meant to be the indicator function of the set E. $\endgroup$
    – Louigi Addario-Berry
    Commented Nov 19, 2010 at 18:05
  • $\begingroup$ 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. $\endgroup$
    – Alex B.
    Commented Nov 20, 2010 at 5:01
  • $\begingroup$ 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? $\endgroup$
    – Louigi Addario-Berry
    Commented Nov 20, 2010 at 10:44

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