Singular Values of Trilinear Forms

Singular Values of Trilinear Forms

Let T : H1 H2 H3 ! C be a trilinear form where H1, H2, H3 are separable Hilbertspaces. In the hypothesis that at least two of the three spaces are nite dimensional weshow that the norm square = kTk2 is a root of a certain algebraic equation, usually ofvery high degree, which we baptize the milleniar equation, because it is an analogue ofthe secular equation in the bilinear c...

Singular Lines of Trilinear Forms

We prove that an alternating e-form on a vector space over a quasialgebraically closed field always has a singular (e − 1)-dimensional subspace, provided that the dimension of the space is strictly greater than e. Here an (e−1)-dimensional subspace is called singular if pairing it with the e-form yields zero. By the theorem of Chevalley and Warning our result applies in particular to finite bas...

Quaternary binary trilinear forms

Dear friends, In 1997 or 1998 I began to write a paper [20], trying to extend the known description of binary forms of norm unity, in the real or complex cases, to the case of quaternions, but I never finished it. In 2002, I resumed my work but then I was unable to understand what I had done, so I made a fresh start with somewhat different assumptions. Now I have realized that this was a gross ...

Singular values of convex functions of matrices

‎Let $A_{i},B_{i},X_{i},i=1,dots,m,$ be $n$-by-$n$ matrices such that $‎sum_{i=1}^{m}leftvert A_{i}rightvert ^{2}$ and $‎sum_{i=1}^{m}leftvert B_{i}rightvert ^{2}$  are nonzero matrices and each $X_{i}$ is‎ ‎positive semidefinite‎. ‎It is shown that if $f$ is a nonnegative increasing ‎convex function on $left[ 0,infty right) $ satisfying $fleft( 0right)‎ ‎=0 $‎, ‎then  $$‎2s_{j}left( fleft( fra...

On Canonical Binary Trilinear Forms