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Re: Matrix Inverse Issue

• To: mathgroup at smc.vnet.net
• Subject: [mg80600] Re: [mg80556] Matrix Inverse Issue
• From: Murray Eisenberg <murray at math.umass.edu>
• Date: Sun, 26 Aug 2007 23:10:38 -0400 (EDT)
• Organization: Mathematics & Statistics, Univ. of Mass./Amherst
• References: <200708260700.DAA03587@smc.vnet.net>
• Reply-to: murray at math.umass.edu

If A and B are matrices of the same size, then A B is NOT the matrix 
product of A and B; rather, it is the matrix of that same size that you 
get by multiplying corresponding elements of A and B.  Multiplication of 
numbers is Listable -- it extends to lists, and lists of lists, ..., 
when their dimensions permit it.

The matrix product of two compatibly-sized matrices A and B is A.B, 
shorthand for Dot[A,B].

So Mathematica gave you precisely what you asked for.  What you wanted 
-- and it could not read your mind -- is:

   A = Table[If[j==i, 1-i/10, If[j==i+1, i/10, 0]], {i, 9}, {j, 9}];
   A.Inverse[A]==IdentityMatrix[9]
True

(I used the test for equality with identity matrix there to spare 
showing the multi-line output of the identity matrix.)

Brian Gladman wrote:
> I am having a problem with the matrix inverse for the following matrix in 
> Mathematica 6:
> 
> A = Table[If[j == i, 1 - i/10, If[j == i + 1, i/10, 0]], {i, 9}, {j, 9}]
> 
> {{9/10, 1/10, 0, 0, 0, 0, 0, 0, 0}, {0, 4/5, 1/5, 0, 0, 0, 0, 0,
>   0}, {0, 0, 7/10, 3/10, 0, 0, 0, 0, 0}, {0, 0, 0, 3/5, 2/5, 0, 0, 0,
>   0}, {0, 0, 0, 0, 1/2, 1/2, 0, 0, 0}, {0, 0, 0, 0, 0, 2/5, 3/5, 0,
>   0}, {0, 0, 0, 0, 0, 0, 3/10, 7/10, 0}, {0, 0, 0, 0, 0, 0, 0, 1/5, 4/
>   5}, {0, 0, 0, 0, 0, 0, 0, 0, 1/10}}
> 
> A Inverse[A]
> 
> {{1, -1/72, 0, 0, 0, 0, 0, 0, 0}, {0, 1, -1/14, 0, 0, 0, 0, 0, 0}, {0,
>    0, 1, -3/14, 0, 0, 0, 0, 0}, {0, 0, 0, 1, -8/15, 0, 0, 0, 0}, {0,
>   0, 0, 0, 1, -5/4, 0, 0, 0}, {0, 0, 0, 0, 0, 1, -3, 0, 0}, {0, 0, 0,
>   0, 0, 0, 1, -49/6, 0}, {0, 0, 0, 0, 0, 0, 0, 1, -32}, {0, 0, 0, 0,
>   0, 0, 0, 0, 1}}
> 
> I expected the identity matrix here but I get extra terms.
> 
> Am I doing something wrong?
> 
>    Brian Gladman
> 
> 

-- 
Murray Eisenberg                     murray at math.umass.edu
Mathematics & Statistics Dept.
Lederle Graduate Research Tower      phone 413 549-1020 (H)
University of Massachusetts                413 545-2859 (W)
710 North Pleasant Street            fax   413 545-1801
Amherst, MA 01003-9305

• References:
  • Matrix Inverse Issue
    • From: "Brian Gladman" <brg@nowhere.org>