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JordanDecomposition trouble

• To: mathgroup@smc.vnet.net
• Subject: [mg10604] JordanDecomposition trouble
• From: Josip Loncaric <josip@icase.edu>
• Date: Mon, 26 Jan 1998 04:42:35 -0500
• Organization: ICASE

Let {s,j} denote Jordan decomposition of the following exact matrix:

m = {{0,1,0,1},{0,0,0,0},{0,1,0,1},{0,0,0,0}} {s,j} =
JordanDecomposition[m]

We should have m == s.j.Inverse[s], but in Mathematica 3.01 this is not
the case.
Instead, we get a rather different result:

In[3]:= s.j.Inverse[s]
Out[3]= {{0,1,0,2},{0,0,0,0},{0,1,0,2},{0,0,0,0}}

The problem is not in matrix inversion since s.Inverse[s] returns
identity.  The source of trouble appears to be in the
JordanDecomposition result.

Jordan decomposition notorious for creating numerical difficulties, but
exact matrices should not lead to such failures.

The situation gets even worse for larger versions of m.  Here is what
happens:

size  = 8;
m     = Table[If[OddQ[i] && EvenQ[j], 1, 0],{i,size},{j,size}]

Out[6]= {{0,1,0,1,0,1,0,1},{0,0,0,0,0,0,0,0},
         {0,1,0,1,0,1,0,1},{0,0,0,0,0,0,0,0},
         {0,1,0,1,0,1,0,1},{0,0,0,0,0,0,0,0},
         {0,1,0,1,0,1,0,1},{0,0,0,0,0,0,0,0}}

{s,j} = JordanDecomposition[m];
s.j.Inverse[s]

Out[7]=  {{0,1,0,3,0,15,0,90},
          {0,0,0,0,0,0, 0,0},
          {0,1,0,3,0,15,0,90},
          {0,0,0,0,0,0, 0,0},
          {0,1,0,3,0,15,0,90},
          {0,0,0,0,0,0, 0,0},
          {0,1,0,3,0,15,0,90},
          {0,0,0,0,0,0, 0,0}}

This is not even close to m, although it should be exactly m.

-- 
Dr. Josip Loncaric, Senior Staff Scientist ICASE, M/S 403, NASA Langley
Research Center, Hampton, VA 23681-0001 Phone: (757) 864-2192          
mailto:josip@icase.edu Fax:   (757) 864-6134                  
http://www.icase.edu/~josip/

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  • Re: JordanDecomposition trouble
    • From: Daniel Lichtblau <danl@wolfram.com>