       Re: MathSource Notebook Smith Normal Form Error - How to Correct?

• To: mathgroup at smc.vnet.net
• Subject: [mg131557] Re: MathSource Notebook Smith Normal Form Error - How to Correct?
• From: daniel.lichtblau0 at gmail.com
• Date: Thu, 29 Aug 2013 02:23:03 -0400 (EDT)
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```On Tuesday, August 27, 2013 1:59:04 AM UTC-5, amzoti wrote:
> Hi
>
> I am trying to do Rational Canonical Forms for matrices and there are no built-in things for this and Smith Normal Forms.
>
> I want to calculate the rational canonical matrix and the P such that A = PRP^(-1). This is a generalized variant of the Jordan Form (which I know how to do).
>
> I found this notebook on Math Source from V6: http://library.wolfram.com/infocenter/MathSource/7081/
>
> It would be great if this package could show how the invariant canonical decomposition of [xI - A] was done (row and column operations).
>
> When I download this and try using it, it does not work and I cannot figure out why.
>
> Is there a way to correct this package so it works in Mathematica Version 9?
>
> Did I miss some other command in Mathematica V9 that can do all of this stuff?
>
> Thanks

What did you actually try? I show a simple example below that appears to give a plausible result (but it's late and I'm not going to check it).

In:= PolynomialSmithForm[x*IdentityMatrix - {{1,2,3},{3,-1,6},{2,7,1}},x]

Out= {{{1, 0, 0}, {0, 1, 0}, {0,
0, -44 - 55 x - x^2 + x^3}}, {{{0, -(1/3), 0}, {4/167, 14/
167, -(23/167) + (2 x)/167}, {23 + 2 x, -3 + 7 x, -7 + x^2}}, {{1,
1/3 + x/3, -(929/501) + (98 x)/501 + (23 x^2)/501 - (2 x^3)/
501}, {0, 1, 73/167 + (25 x)/167 - (2 x^2)/167}, {0, 0, 1}}}}

Mathematica 9 does not have this type of function. I expect there to be a well hidden way to do it in version 10. That is to say, it will not be in System` context. ControlTheory`, perhaps. That's for the polynomial version. I need to check whether we will also have the integer version there.

Daniel Lichtblau
Wolfram Research

```

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