 # Glas :

## Re: [glas] norms and inner products for vectors of matrices

From: Karl Meerbergen (Karl.Meerbergen_at_[hidden])
Date: 2006-01-09 07:12:56

On Monday 09 January 2006 12:55, Neal Becker wrote:
> > Certainly, at least for the norm. For the inner product, should it be
> > trans(v[i]) * w[i]
> > or
> > v[i] * w[i]
> > I guess the former makes sense if one thinks of vectors that have a block
> > structure (in your diction, each block would be a n times 1 matrix).
>
> I think you want v[i] * conj (w[i]).

Note that the dot product does not have a conjugate transpose.
In a Linear Algebra notation, the (Euclidean) inner product would be
herm(w)*v
which is translated into
sum_i conj(w[i])*v[i]
when w[i] and v[i] are scalars.

Alternative notations could be
trans(conj(w))*v (very unusual)
herm_trans(w)*v

When w and v's value_type's are collections themselves, we would have:
herm(w)*v = sum_i herm(w[i])*v[i]
Assuming that herm(x)=conj(x) for a scalar, this also holds for vectors of
scalars.

Karl