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| newbie: how to find minimal polynomial of a matrix? https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1898 |
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| Author: | alhutch [ Mon Jan 03, 2011 11:20 pm ] |
| Post subject: | newbie: how to find minimal polynomial of a matrix? |
Thank you for Singular. I want to do what seems a simple job: to find the minimal polynomial of a square matrix. The matrix entries are polynomials in the ring Q[x], so the minimal polynomial will also be in this ring. The algorithmic complications seem to be - although all the entities involved are in commutative rings, the calculations of powers of the original matrix are made in the ring of all such matrices over Q[x], which is not commutative; - the tools of Singular seem designed (for good reason) to hide the information I want. They tell the user whether an element is in an ideal, but not (obviously) how the element can be expressed in terms of the ideal's generators. Any guidance welcome. |
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| Author: | malex [ Tue Jan 04, 2011 8:52 pm ] |
| Post subject: | Re: newbie: how to find minimal polynomial of a matrix? |
1. don't you need to adjoin another commutative variable to your ring (e.g. Q[x] -> Q[x][T])? 2. matrix multiplication is generally non-commutative, no? 3. one uses 'lift' for "how the element can be expressed in terms of the ideal's generators" 4. i'd suggest you to start with characteristic polynomial (of M) as follows: a. compute D = det(M) in our original ring, b. switch to enriched ring (e.g. new variable T), and imap D and M there c. compute det(D * E * T - M) (where E is the identity matrix) it can be used for testing and checking your results... ps: Happy New Year |
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| Author: | alhutch [ Wed Jan 05, 2011 11:16 am ] |
| Post subject: | Re: newbie: how to find minimal polynomial of a matrix? |
Thanks, malek. This makes things much clearer. Alan. |
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