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From: moren.neron on 9 Aug 2010 09:25
Call a mxn matrix "rectangular unitary" if each of its lines is a unit complex vector and each two of its rows is orthogonal. Denote V^t the transpose of a matrix V.

For a square matrix W, let S(W) be the sum of all its entries (obs. and not the sum of the absolute values
of the entries).

I'm interested in providing the best upper bound for
the following number, where the maximum is taken over
all rectangular unitary matrices U,V,

M_n = Max_{U,V} Sum( U*V^t )


Let U,V be two nxm matrices whose rows are complex unit vectors and such that any two rows of U are orthogonal and any two rows of V are orthogonal.

A trivial lower bound is M_n = n, corresponding to the case that U=V, and a trivial upper bound is n^2, corresponding to the (impossible) case where all inner
products <u_i,v_j>=1, where u_i, and v_j are the lines of U and V respectively.

thank you very much for the attention,

moren
From: Chip Eastham on 9 Aug 2010 15:04
On Aug 9, 1:25 pm, "moren.neron" <moren.ne...(a)gmail.com> wrote:
> Call a mxn matrix "rectangular unitary" if each of its lines is a unit complex vector and each two of its rows is orthogonal. Denote V^t the transpose of a matrix V.
>
> For a square matrix W, let S(W) be the sum of all its entries (obs. and not the sum of the absolute values
> of the entries).
>
> I'm interested in providing the best upper bound for
> the following number, where the maximum is taken over
> all rectangular unitary matrices U,V,
>
>   M_n = Max_{U,V} Sum( U*V^t )
>
> Let U,V be two nxm matrices whose rows are complex unit vectors and such that any two rows of U are orthogonal and any two rows of V are orthogonal.
>
> A trivial lower bound is M_n = n, corresponding to the case that U=V, and a trivial upper bound is n^2, corresponding to the (impossible) case where all inner
> products <u_i,v_j>=1, where u_i, and v_j are the lines of U and V respectively.
>
> thank you very much for the attention,

Hi, moren:

I'm confused by the introduction of a general
notion of "rectangular unitary" complex matrices
of size mxn, but no reference to m (presumably
the number of "lines" or rows in each of U,V)
in formulating the maximum over U,V. Note you
later described U,V as being nxm matrices.

Won't the bounds depend on the relationship
between m and n? In particular if m < n vs.
m > n, doesn't that effect what you describe
as a "trivial lower bound" (on M_n)?

regards, chip
From: Gerry Myerson on 10 Aug 2010 00:04
In article
<b9633bf8-73fe-4228-b302-56f470237087(a)x25g2000yqj.googlegroups.com>,
Chip Eastham <hardmath(a)gmail.com> wrote:

> On Aug 9, 1:25�pm, "moren.neron" <moren.ne...(a)gmail.com> wrote:
> > Call a mxn matrix "rectangular unitary" if each of its lines is a unit
> > complex vector and each two of its rows is orthogonal. Denote V^t the
> > transpose of a matrix V.
> >
> > For a square matrix W, let S(W) be the sum of all its entries (obs. and not
> > the sum of the absolute values
> > of the entries).
> >
> > I'm interested in providing the best upper bound for
> > the following number, where the maximum is taken over
> > all rectangular unitary matrices U,V,
> >
> > � M n = Max {U,V} Sum( U*V^t )
> >
> > Let U,V be two nxm matrices whose rows are complex unit vectors and such
> > that any two rows of U are orthogonal and any two rows of V are orthogonal.
> >
> > A trivial lower bound is M n = n, corresponding to the case that U=V, and a
> > trivial upper bound is n^2, corresponding to the (impossible) case where
> > all inner
> > products <u i,v j>=1, where u i, and v j are the lines of U and V
> > respectively.
> >
> > thank you very much for the attention,
>
> Hi, moren:
>
> I'm confused by the introduction of a general
> notion of "rectangular unitary" complex matrices
> of size mxn, but no reference to m (presumably
> the number of "lines" or rows in each of U,V)
> in formulating the maximum over U,V. Note you
> later described U,V as being nxm matrices.
>
> Won't the bounds depend on the relationship
> between m and n? In particular if m < n vs.
> m > n, doesn't that effect what you describe
> as a "trivial lower bound" (on M n)?

In order to have "any two rows of U are orthogonal"
you have to have number of rows no more than number
of columns, so I think we can take it for granted that
the n-by-m matrices have n at most m.

It is not clear to me that the bounds wanted
are independent of m, but neither is it clear to me
that they aren't.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
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