Let $V$ be a finite dimensional vector space, with some choice of basis $\{e_i\}_{i \in I}$. With respect to an idempotent bijection $B:I \to I$, define a bilinear form by $$ g = \sum_{i=1}^N \lambda_{i} e^i \otimes e^{B(j)}, $$ where $\{e^i\}_{i \in I}$ is the dual basis of $V^*$. Moreover, assume that $\lambda_i \neq \lambda_{I(i)}$, implying that $g$ is not a symmetric form. I would like to know if such a $g$ has a name, and if anyone has previously considered such objects.
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$\begingroup$ Do you mean $B(i)$ rather than $I(j)$ in the superscript? And is $I=\lbrace 1, \dots, N \rbrace$? $\endgroup$ – MTS Jul 26 '13 at 18:29

$\begingroup$ . . . and yes $I=\{1,\ldots, N\}$. $\endgroup$ – Milan Bernolak Jul 26 '13 at 18:43
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You may find the following link helpful: http://en.wikipedia.org/wiki/Generalized_permutation_matrix