Origin of decomposition theorem

Question

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).

Answer 1

I do think this is a difficult question for a few reasons. What are we to accept as decomposition, because some of these ideas have been reconceptualized over time. Let me illustrate the difficulty by proposing one possible answer:

Daniel Bernoulli proposed that the solution of motion of a string under tension can be described as superposition of oscillations. Today we would say that this is a "Fourier decomposition" and it can indeed be thought of an eigen-decomposition over an infinite-dimensional vector space. But are we granting D. Bernoulli's work too much of a post-hoc lens with this example?

• Bernoulli, D. (1753/5) Réflexions et éclaircissemens sur les nouvelles vibrations des cordes exposées dans les mémotres de l’académie de 1747 & 1748,” Hist. acad. Berlin 9, pp. 147—172.

Many decomposition theorems in linear algebra developed throughout the 19th century (see Jordan normal form for example), but the examples that follow your modern notation and their deeper structural understanding with respect to algebraic classification is work of the early 20th century.

But if we think of decomposition more loosely, for example in terms of decomposing some whole into sub-pieces, then basic counting already provides examples (how many equal sized integers can a given integer be "decomposed" into? In this sense the historical roots of decompositions must be ancient.