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Topic: Symbolic Jordan Decomposition
Replies: 5   Last Post: Jan 9, 2012 2:33 PM

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Leslaw Bieniasz

Posts: 173
Registered: 12/13/04
Re: Symbolic Jordan Decomposition
Posted: Jan 9, 2012 5:00 AM
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On Mon, 9 Jan 2012, wrote:

> Leslaw Bieniasz schrieb:

>> Some computer algebra programs have built-in functions for computing
>> the Jordan normal form of a matrix. For example, MATHEMATICA has the
>> function JordanDecompose[], MATLAB has Jordan(). Most importantly,
>> these functions can work on matrices given in a symbolic way (matrix
>> entries represented by symbols, not concrete numbers). My question is:
>> what kind of algorithms is implemented in these functions in the
>> symbolic mode of operation? I would appreciate pointers to the
>> relevant mathematical literature describing the algorithms.

> This may be hard to find out if no open-source system can be found with
> the same function.
> The Jordan canonical form of a matrix exposes the eigenvalues on its
> diagonal, but in general these diagonal elements cannot be determined
> symbolically if there are five or more distinct eigenvalues. (I believe
> I read somewhere that the roots of fifth-degree polynomials can always
> be given in terms of hypergeometric functions, however.)

In general no, but there are plenty of examples when analytical roots
of high order polynomials do exist, for sufficiently simple matrices.
These are exactly the cases I am most interested in.

> The procedure for those cases where the eigenvalues of the input matrix
> can be determined symbolically is described in various textbooks, for
> instance in Smirnov III/1,27 and III/2,189-194. I fear that the textbook
> recipes become impractical if one tries to handle five or more distinct
> eigenvalues as RootOf[] objects. Do Mathematica and Matlab return
> anything in such cases?

I am not sure about MATLAB, but I think MATHEMATICA returns correct
eigenvalues symbolically, even for n=5 or more (in cases when such
formulae exist).

In any case, I don't think that the approach based on looking for
eigenvalues is really employed in MATHEMATICA. I have info now
that JordanDecompose[] uses "the method based on matrix presentations",
whatever this may mean. I presume it uses some kind of matrix
transformations to arrive at the Jordan form, and eigenvalues are
obtained as a by-product.

What is that Smirnov III ?


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