Jordan Form

• To: mathgroup at smc.vnet.net
• Subject: [mg26310] Jordan Form
• From: "Matt Herman" <Henayni at hotmail.com>
• Date: Sun, 10 Dec 2000 21:37:56 -0500 (EST)
• Sender: owner-wri-mathgroup at wolfram.com

```Hi,

I have found a bug in JordanDecomposition.

Consider the matrix A
{
{2, 0, 0, 0, 0, 0, 0, 0},
{1, 2, 0, 0, 0, 0, 0, 0},
{-4, 1, 2, 0, 0, 0, 0, 0},
{2, 0, 0, 2, 0, 0, 0, 0},
{-7, 2, 0, 0, 2, 0, 0, 0},
{9, 0, -2, 0, 1, 2, 0, 0},
{-34, 7, 1, -2, -1, 1, 2, 0},
{145, -17, -16, 3, 9, -2, 0, 3}
}.

Then, Mathematica gives
In[3]:=
JordanDecomposition[A][[2]]
Out[3]=
{
{2, 1, 0, 0, 0, 0, 0, 0},
{0, 2, 0, 0, 0, 0, 0, 0},
{0, 0, 2, 1, 0, 0, 0, 0},
{0, 0, 0, 2, 0, 0, 0, 0},
{0, 0, 0, 0, 2, 1, 0, 0},
{0, 0, 0, 0, 0, 2, 1, 0},
{0, 0, 0, 0, 0, 0, 2, 0},
{0, 0, 0, 0, 0, 0, 0, 3}
}

Which is J(2,2)+J(2,2)+J(2,3)+J(3,1), where J(c,n) is the nxn matrix
with c's on the diagonal, and ones on the upper (or lower)
"subdiagonal".

However, if you do it by hand (i.e. computing nullspaces, etc..), you
get that the JCF is J(2,3)+J(2,3)+J(2,1)+J(3,1), as confirmed by other
systems. But the Jordan form is unique, so we have a problem.

Any ideas (and no, this is not a matter of not entering the matrix
correctly)?

```

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