How Is the Primary Decomposition Theorem Known in Various Mathematical Texts?

Treadstone 71
Messages
275
Reaction score
0
"Let m(x) be the minimal polynomial of T:V\rightarrow V, \dim V<\infty such that m(x)=m_1(x)m_2(x) where gcd(m_1,m_2)=1, then there exists T-invariant subspaces V_1, V_2 such that V=V_1\oplus V_2."

What other names is this thoerem called? It was given to me as the "primary decomposition theorem" but it's neither in my book nor in mathworld or wikipedia.
 
Last edited:
Physics news on Phys.org
look on my webpage, http://www.math.uga.edu/~roy/ and download my linear algebra notes. this is proved there.

the proof is based on the linear combination proeprty for gcd's, i.e. you can write 1 as a linear combination of two polynbomilas with gcd = 1, so if 1 = pm1 + qm2, then plugging in T for X, we get id = p(T)m1q(T) +q(T)m2(T). so we have decomposed the identity map into two direct sum components. this decompsoes V accordingly. i.e. V1 = ker pm1, and V2 = ker qm2. is that right? (i am a little under the weather at the moment.)
 
Yup, this is it. Thanks.
 
my pleasure. we live in the hope of being of service, and occsionally this occurs.
 

Thread '##(A/\mathfrak{a})_{\mathfrak{p}/\mathfrak{a}}## and its isomorphism?'

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to A_\mathfrak{p}##? But I don't understand what the author meant for the rest of the sentence in mathematical notation: ##2.## In the next statement where the author says: How is ##A\to...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

How Does the Direct Sum Relate to Unique Decomposition in Vector Spaces?
Replies
1
Views
2K
Confusion with non-head-on elastic collisions
Replies
4
Views
5K
Challenge Math Challenge - April 2021
3
Replies
102
Views
10K
Challenge Math Challenge - November 2018
4
Replies
175
Views
25K
Challenge Intermediate Math Challenge - September 2018
Replies
28
Views
6K
Challenge Intermediate Math Challenge - July 2018
Replies
16
Views
6K
Mathematica This Week's Finds in Mathematical Physics (Week 257)
Replies
5
Views
4K

Hot Threads

Recent Insights

Back
Top