# Primary Decomposition Theorem

1. Feb 24, 2006

"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: Feb 24, 2006
2. Feb 24, 2006

### mathwonk

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.)

3. Feb 24, 2006