Hadamard (component-wise) vector product?

[Chris Hecker]

Can someone explain what the Hadamard product is used for in math? It's
definied for two vectors as z_i = x_i * y_i, and I believe it's also
sometimes called the "vector multiply" (Golub and van Loan) or the
"componentwise product," and I think it's denoted in TeX by x \circ y or
something like that (I don't really know TeX).

Basically, I'm looking for a description of what is calculates, in the
same way that I can say the dot product calculates the cos(angle)
between vectors, etc.

I found a bunch of references to it on the web, but never a description
of what it's actually doing with the input vectors. It's mentioned in
Golub and van Loan's Matrix Computations at the beginning, but I don't
know if it's mentioned again, and G&vL is at work and I keep forgetting
to look when I'm at work and it's driving me nuts as I sit here at
home. :)

Thanks!

Chris

[John Hench]

Chris Hecker wrote:

> Can someone explain what the Hadamard product is used for in math?

For vectors, I suppose that the Hadamard product
could be used to build up the column vectors of
the system matrix for a Least Squares Polynomial
Approximation. If F(t) = [ 1, 2, 3, 4]' is your
sample points in time and D(y) = [ 1, 3, 1, -1]'
is your data at the corresponing times in F, and
you want to find the least squares estimate a +
bt + ct^2 = y, you could find the solution by
forming the matrix A, where

A =

[ 1 1 1 ]
[ 1 2 4 ]
[ 1 3 9 ]
[ 1 4 16 ]

and solving Ax = b in the least square sense,
that is, solving A'Ax=A'b, where b = D(y), and
A = [F.^0, F.^1, F.^2].

I've seen two uses as far as matrices go. One
use is may be found in the relationship between
matrices that are derived from various tensor
products: see "Matrix Derivatives" by Gerald
Rogers. I also vaguely remember a paper written
by a Chemical Engineer in which he used the
Hadamard product of two matrices to describe an
array that was important in Chemical Engineering.
The array was computed from two other matrices
thusly: B = A \circ A^{-T}, if I remember
correctly. I suspect I'm wrong about the
details. Could anyone remind me of that result?

Thanks, John

-------------------------------------------------
Dr. J.J. Hench
Dept. of Mathematics, Univ. of Reading, England
Institute of Informatics and Automation, Prague
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