Decomposition of bilinear forms as sums of bounded forms

Bilinear Forms

The geometry of Rn is controlled algebraically by the dot product. We will abstract the dot product on Rn to a bilinear form on a vector space and study algebraic and geometric notions related to bilinear forms (especially the concept of orthogonality in all its manifestations: orthogonal vectors, orthogonal subspaces, and orthogonal bases). Section 1 defines a bilinear form on a vector space a...

Decomposition of quantics in sums of powers of linear forms

Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent use of High-Order Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering problems. In this paper, we focus on the decomposition of a symmetric tensor into a sum of simpler ones, and links ...

Arens regularity of bilinear forms and unital Banach module spaces

Assume that $A$‎, ‎$B$ are Banach algebras and that $m:Atimes Brightarrow B$‎, ‎$m^prime:Atimes Arightarrow B$ are bounded bilinear mappings‎. ‎We study the relationships between Arens regularity of $m$‎, ‎$m^prime$ and the Banach algebras $A$‎, ‎$B$‎. ‎For a Banach $A‎$‎-bimodule $B$‎, ‎we show that $B$ factors with respect to $A$ if and only if $B^{**}$ is unital as an $A^{**}‎$‎-module‎. ‎Le...

Apolarity of Trilinear Forms and Pencils of Bilinear Forms*

Association Schemes of Quadratic Forms and Symmetric Bilinear Forms

Let Xn and Yn be the sets of quadratic forms and symmetric bilinear forms on an n-dimensional vector space V over Fq , respectively. The orbits of GLn(Fq ) on Xn × Xn define an association scheme Qua(n, q). The orbits of GLn(Fq ) on Yn × Yn also define an association scheme Sym(n, q). Our main results are: Qua(n, q) and Sym(n, q) are formally dual. When q is odd, Qua(n, q) and Sym(n, q) are iso...