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Let $V=(V, \langle\; , \; \rangle)$ denote an inner product space. Then the concept "Gramian matrix" is the function \begin{align*}V^n &\rightarrow \mathbb{R}^{n \times n} \\x &\mapsto \sum_{i,j=1}^n \langle x_i,x_j\rangle e_i e_j^\top.\end{align*}

Suppose we're interested in the two-argument version of this, namely \begin{align*}V^n \times V^n &\rightarrow \mathbb{R}^{n \times n}, \\ (x,y) &\mapsto \sum_{i,j=1}^n \langle x_i,y_j\rangle e_i e_j^\top. \end{align*}

The determinant of this matrix shows up in the definition of the dot product of multivectors.

Question. Does this function $V^n \times V^n \rightarrow \mathbb{R}^{n \times n}$ have a name and/or accepted notation, like "two-argument Gram matrix" or something like that?

Also, is there an accepted notation for it?

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  • $\begingroup$ Where did you find this definition of gramian matrix (the ordinary "single-argument")? $x_i$ and $x_j$ are components of $x$? Then what is it meant by $\langle x_i,x_j\rangle$? $\endgroup$ – trying Jul 27 '17 at 7:37
  • $\begingroup$ @trying, that's the inner product. You can think of this as $x_i \bullet y_j$. Look up "inner product space." Note that, since $x$ is a list of vectors, hence $x_i$ is a vector, not a a scalar. $\endgroup$ – goblin Jul 27 '17 at 8:42
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The determinant in the question you cited is called cross-gramian. See here.

Correspondingly the matrix is called cross-gramian operator or cross-gram matrix as is reported in "Bridging erasures and the infrastructure of frames" a chapter of R. Balan et al. (eds.) - Excursions in Harmonic Analysis, Volume 4

There also the notation used are these: $$Gr(x,y)=(\langle x_i,y_j\rangle)_{i,j} = \sum_{i,j=1}^n \langle x_i,y_j\rangle e_i \otimes e_j$$

I think that $x^Ty$ could conveniently be used, generalizing the notation of the gramian matrix given in this article.

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