primary decomposition theorem
This is an important theorem in linear algebra. It states the following:
Let k be a field, V a vector space
over k, dimV=n, and T:V→V a linear operator, such that its minimal polynomial (or its annihilator polynomial) is mT, which decomposes in k[X] into irreducible factors as mT=pα11…pαrr. Then,
-
1.
V=⊕ri=1ker(pαii(T))
-
2.
ker(pαii(T)) is T-invariant for every i
-
3.
If Ti is the restriction
of T to ker(pαii(T)), then mTi=pαii
This is a consequence of a more general theorem: Let V, T be as above, and f∈k[X] such that f(T)=0, with f=f1…fr and (fi,fj)=1 if i≠j, then
-
1.
V=⊕ri=1ker(fi(T))
-
2.
ker(fi(T)) is T-invariant for every i
To illustrate its importance, the primary decomposition theorem, together with the cyclic decomposition theorem, imply the existence and uniqueness of the Jordan canonical form
.
Title | primary decomposition theorem |
---|---|
Canonical name | PrimaryDecompositionTheorem |
Date of creation | 2013-03-22 14:15:30 |
Last modified on | 2013-03-22 14:15:30 |
Owner | gumau (3545) |
Last modified by | gumau (3545) |
Numerical id | 8 |
Author | gumau (3545) |
Entry type | Theorem |
Classification | msc 15A04 |