proof of Cayley-Hamilton theorem in a commutative ring
Let be a commutative ring with identity and let be an order matrix
with elements from .
For example, if is
then we can also associate with the following polynomial having matrix coefficents:
In this way we have a mapping which is an isomorphism of the rings and .
Now let and
consider the characteristic polynomial of : , which is a monic
polynomial
of degree with coefficients in .
Using a property of the adjugate matrix we have
Now view this as an equation in . It says that is a left factor of . So by the factor theorem, the left hand value of at is 0. The coefficients of have the form , for , so they commute with . Therefore right and left hand values are the same.
References
-
1
Malcom F. Smiley. Algebra
of Matrices. Allyn and Bacon, Inc., 1965. Boston, Mass.
Title | proof of Cayley-Hamilton theorem![]() |
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Canonical name | ProofOfCayleyHamiltonTheoremInACommutativeRing |
Date of creation | 2013-03-22 16:03:16 |
Last modified on | 2013-03-22 16:03:16 |
Owner | Mathprof (13753) |
Last modified by | Mathprof (13753) |
Numerical id | 11 |
Author | Mathprof (13753) |
Entry type | Proof |
Classification | msc 15A18 |
Classification | msc 15A15 |