# Origin of decomposition theorem

There are many different types of decomposition theorem in linear algebra. For example, there are primary decomposition theorem, cyclic decomposition theorem, etc. But I became curious about the history of the decomposition theorem itself and what the first decomposition theorem might have been. What was the decomposition theorem that first appeared, and what is the background or history of its appearance?

Primary decomposition theorem

Suppose that $$m_T (x) = f_1(x)^{m_1}f_2(x)^{m_2} . . . f_k(x)^{m_k}$$, where $$f_1, f_2, . . . , f_k$$ are distinct monic irreducible polynomials over $$F$$. Then $$V = V_1 ⊕ V_2 ⊕ · · · ⊕ V_k$$, where $$V_1, V_2, . . . , V_k$$ are $$T$$-invariant subspaces and the minimal polynomial of $$T|V_i$$ is $$f_i^{m_i}$$ for $$1≤i≤k$$.

Cyclic decomposition theorem

$$T$$ in $$L(V,V)$$, $$V_n$$-dim v.s. $$W_0$$ proper $$T$$-admissible subspace. Then there exist nonzero $$a_1,…,a_r$$ in $$V$$ and respective $$T$$-annihilators $$p_1,…,p_r$$ such that

(i) $$V=W_0 ⊕Z(a_1;T) ⊕… ⊕Z(a_r;T)$$

(ii) $$p_k$$ divides $$p_{k-1}, k=2,..,r$$.

Furthermore, $$r, p_1,..,p_r$$ uniquely determined by (i),(ii) and $$a_i ≠0$$. ($$a_i$$ are not nec. unique).

• Could you state the theorem you are asking about? Commented Jul 6 at 15:10
• @GeraldEdgar Hello, Thank you for your comment! I edited it! Commented Jul 6 at 21:25
• You didn't start with... "Let $V$ be a vector space. Let $T$ be a linear transformation on $V$. Let $m_T(x)$ be ..." and so on Commented Jul 7 at 0:39