Literature about the “Spaltungssatz”

linear algebraminimal-polynomialsreference-request

In the german book "Lineare Algebra" by Theo de Jong, in section 6.7 the so called "Spaltungssatz" is presented (literally: splitting theorem).

It says: Let $m_\varphi = p \cdot q$ be the minimal polynomial of a linear endomorphism $\varphi : V \to V$ such that the polynomials $p, q$ are coprime (i.e. $\gcd(p,q)=1$). Then:

  • $\ker(p(\varphi)) = \text{im}(q(\varphi))$
  • $\ker(q(\varphi)) = \text{im}(p(\varphi))$
  • the above two spaces are non-trivial and $\varphi$-invariant (i.e. they satisfy $\varphi(\ker(p(\varphi))) \subseteq \ker(p(\varphi))$ and analogously for the other)
  • They form a direct sum $V = \ker(p(\varphi)) \oplus \text{im}(p(\varphi))= \ker(q(\varphi)) \oplus \text{im}(q(\varphi))$

Is there any other source for this theorem or an english name for it? I only have parts of the above book available, so I can't check the bibliography. And it does not occur in the other linear algebra books I checked.

Edit:
I found related StackExchange questions, concerning similar properties of the minimal polynomial.

The first mentions the additional property that $\varphi|_{\ker(p(\varphi))}$ has minimal polynomial $p$.
The second mentions the book of Hoffmann and Kunze, in which I could find a “primary decomposition theorem” which is much stronger than the Spaltungssatz of de Jong and gives more insight into the data contained in the minimal polynomial.

Best Answer

The primary decomposition theorem in the book of Hoffmann and Kunze seems to be a fitting answer. Most of it follows by applying the Spaltungssatz of de Jong inductively. For reference I reproduce/reformulate its statement: (p. 220, Chapter 6, Theorem 12 & Corollary)

Let $\varphi : V \to V$ be a linear endomorphism. Decompose its minimal polynomial into irreducible factors (this can be done over any field). I.e. let $p_1, \dots, p_k$ be distinct irreducible monic polynomials and $n_1, \dots, n_k$ be positive integers such that $m_\varphi = \prod_{i=1}^k p_i^{n_i}$. Further, define $W_i := \ker (p_i(\varphi))$. Then the following statements hold:

  • The spaces $W_i$ are non-trivial and $\varphi$-invariant.
  • For each $i$, the restriction of $\varphi$ to $W_i$ has minimal polynomial $p_i^{n_i}$. Consequently, $\dim(W_i) \ge n_i \cdot \deg (p_i)$.
  • The spaces $W_i$ form a direct sum $V = \bigoplus_{i=1}^k W_i$. Call the projections associated with this direct sum $\pi_1, \dots, \pi_k$.
  • For each $i$ there is a polynomial $h_i$ such that $h_i(\varphi) =\pi_i$.
  • If another linear endomorphism $\psi : V\to V$ commutes with $\varphi$, then it commutes with each $\pi_i$ and each $W_i$ is $\psi$-invariant.
  • All $\varphi$-invariant subspaces can be written as direct sum of some of the $W_i$.

Parts of this can be expressed in matrix form. There exists a basis of $V$, such that the matrix of $\varphi$ wrt. this basis has block-diagonal form with non-empty blocks $A_1, \dots, A_k$ such that each $A_i$ has minimal polynomial $p_i^{n_i}$. The size of $A_i$ is at least $n_i \cdot \deg(p_i)$.

For the proof, Hoffmann and Kunze first construct the polynomials $h_i$ and derive the rest from there. The statements above which are not included in Hoffmann and Kunze are relatively easy corollaries of the others. All in all, this theorem produces a "best possible" block-diagonal form, without imposing restrictions on the underlying field. From another point of view, it characterises the relation between invariant subspaces and factors of the minimal polynomial.

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