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Subject: Re: calculating the inverse efficiently (not for solving equations :-) )
Date: Wed, 2 Apr 2008 01:50:03 +0000 (UTC)
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"Tim Davis" <davis@cise.ufl.edu> wrote in message <fsukql$df3
$1@fred.mathworks.com>...
> If you ever find yourself multiplying by the inverse, then
> you know one thing for certain.  You know, for sure, that
> you don't know what you're doing.  One needs never, ever,
> ever multiply by the inverse.  Multiplying by the inverse is
> mathematically equivalent to solving a system of linear
> equations, but it is an awful computational replacement for
> the latter.
> ...........
---------
  Tim, it is best not to make claims that can be misunderstood, as in: "If you 
ever find yourself multiplying by the inverse, then you know one thing for 
certain.  You know, for sure, that you don't know what you're doing.  One 
needs never, ever, ever multiply by the inverse."  There is a Murphy's Law-like 
principle that states that someone, sometime, somewhere is surely going to 
come up with an example for which multiplying by the inverse of a square 
matrix is the very, very best way of carrying out a given task.

  How about this one for starters.  We are given a non-singular 3 x 3 square 
matrix M that transforms 3 x 1 vectors, x, to 3 x 1 vectors, y, in 3D space.  
We are also given a point cloud of 1,000,000 different points of y-values and 
are to determine the x-point cloud they came from by way of multiplication 
by M.  I claim the very best way to do this is to determine, once and for all, by 
whatever means are necessary, fair or foul, the explicit, full inverse of M, and 
then, having accomplished that despicable operation, to then proceed with 
the 1,000,000 matrix multiplications of it by y-points necessary to determine 
all the x-points.

  At this juncture, I can envision you claiming, "Foul, foul, that isn't what I 
meant in my statement!"  But that is exactly the point of my comment here.  
Your extreme statement could easily be interpreted in just such a manner.  It 
is always best to carefully quantify such statements so as not to allow of such 
misinterpretations.

Roger Stafford