Invariant subspaces and the primary decomposition

Intuition Beginner

A linear map moves the whole space around at once, and that can be hard to picture all together. The trick is to look for pieces of the space that the map keeps to themselves. A direction, a plane, or a larger slab is kept to itself by the map if every vector inside it lands back inside it. Such a piece is called an invariant subspace. Inside one, the map is its own self-contained transformation, and you can study it without watching what happens everywhere else.

Eigenvectors are the smallest example. An eigenvector sits on a line that the map only stretches, so the line is kept to itself. The bigger idea is that you can often slice the whole space into several such pieces that the map never mixes, and then the complicated map becomes a stack of simpler maps, one per piece. The matrix in matching coordinates falls into separate blocks along the diagonal, with empty corners, because no piece feeds into another.

The primary decomposition is the cleanest way to do this slicing. It groups directions by which eigenvalue, or which repeated-root factor, controls them. Each group is a slab the map keeps to itself, and the groups together fill the whole space. You break one hard problem into a handful of easy ones, each tied to a single eigenvalue.

Visual Beginner

The picture shows a space split into two slabs that a linear map never mixes. On the left, a transformation acts on all of three-dimensional space at once, arrows pointing every which way. On the right, the same space is drawn as a horizontal plane stacked with a vertical line through it; the map sends plane-vectors back into the plane and line-vectors back into the line, so the two pieces stay separate forever.

Two facts are visible. Because each slab is kept to itself, the matrix in coordinates adapted to the slabs has its non-zero entries gathered into square blocks on the diagonal, with zeros in the corners. And because the slabs together fill the whole space, studying the map on each slab separately tells you everything about the map.

Worked example Beginner

Take the three-by-three matrix

$$A=\left(\begin{array}{ccc}2& 0& 0\\ 0& 4& 1\\ 0& 0& 4\end{array}\right).$$

Step 1. Read off the eigenvalues from the diagonal of this upper-triangular matrix: $2$ and $4$.

Step 2. Find the slab for the eigenvalue $2$. Solve $Av=2v$. The vector $(1,0,0)$ works, because $A(1,0,0)=(2,0,0)=2(1,0,0)$. So the line through $(1,0,0)$ is kept to itself, and on it the map just multiplies by $2$.

Step 3. Find the slab for the eigenvalue $4$. Look at the second and third coordinates, where the active block is $\left(\begin{array}{cc}4& 1\\ 0& 4\end{array}\right)$. The vectors $(0,1,0)$ and $(0,0,1)$ span a plane. Check it is kept to itself: $A(0,1,0)=(0,4,0)$ stays in the plane, and $A(0,0,1)=(0,1,4)$ stays in the plane. So this plane is a second slab.

Step 4. Confirm the two slabs fill the space and do not overlap. The line and the plane together span all three coordinates, and they share only the zero vector, so every vector splits uniquely into a line-part plus a plane-part.

Step 5. See the block structure. In the basis $(1,0,0),(0,1,0),(0,0,1)$, the matrix is already block-diagonal: a one-by-one block holding $2$, and a two-by-two block holding $\left(\begin{array}{cc}4& 1\\ 0& 4\end{array}\right)$, with zeros in the corners.

What this tells us: the eigenvalue $2$ owns a one-dimensional slab and the eigenvalue $4$ owns a two-dimensional slab. The map never mixes them, so the single three-dimensional problem became one easy problem on a line plus one small problem on a plane.

Check your understanding Beginner

Formal definition Intermediate+

Let $K$ be a field, $V$ a finite-dimensional $K$-vector space, and $T:V\to V$ a linear operator in the sense of 01.01.05. The role of the spectral theorem 01.01.13 in this unit is as the orthogonal model case: when $T$ is normal on an inner-product space, the invariant decomposition below is the orthogonal eigenspace decomposition, and the present unit drops both the inner product and the diagonalisability hypothesis.

Definition (invariant subspace). A subspace $W\subseteq V$ is $T$-invariant if $T(W)\subseteq W$, that is $Tw\in W$ for every $w\in W$. The zero subspace $\{0\}$ and the whole space $V$ are $T$-invariant for every $T$. The kernel $\ker T$, the image $\mathrm{im}T$, each eigenspace $E_{\lambda }=\ker (T-\lambda I)$, and each generalised eigenspace $\ker (T-\lambda I{)}^k$ are $T$-invariant; for the generalised eigenspace, $T$ commutes with $(T-\lambda I{)}^k$, so $T$ carries its kernel into itself.

Induced operators. If $W$ is $T$-invariant, the restriction $T{|}_W:W\to W$, $w\mapsto Tw$, is a well-defined operator on $W$. The operator $T$ also descends to the quotient $V/W$: the induced operator $\overline{T}:V/W\to V/W$, $v+W\mapsto Tv+W$, is well-defined precisely because $T(W)\subseteq W$ makes the rule independent of the coset representative.

Block-triangular and block-diagonal forms. Choose a basis $w_1,\dots ,w_m$ of $W$ and extend to a basis $w_1,\dots ,w_m,u_1,\dots ,u_{n-m}$ of $V$. Because $T(W)\subseteq W$, the matrix of $T$ in this basis is block upper-triangular,

$$[T]=\left(\begin{array}{cc}[T{|}_W]& B\\ 0& [\overline{T}]\end{array}\right),$$

with $[T{|}_W]$ the $m\times m$ matrix of the restriction, $[\overline{T}]$ the $(n-m)\times (n-m)$ matrix of the induced operator on the quotient, and $B$ recording how $T$ pushes the complementary directions back toward $W$. If a complementary invariant subspace $U$ exists — a $T$-invariant $U$ with $V=W\oplus U$ — then choosing the $u_j$ inside $U$ forces $B=0$, and the matrix is block-diagonal,

$$[T]=\left(\begin{array}{cc}[T{|}_W]& 0\\ 0& [T{|}_U]\end{array}\right).$$

The existence of a complementary invariant subspace is the difference between a block-triangular reduction (always available from a single invariant subspace) and a block-diagonal splitting (which decouples the operator into independent pieces).

Definition (annihilating and minimal polynomial). The polynomials $p\in K[t]$ with $p(T)=0$ form a non-zero ideal of $K[t]$; its monic generator $m_T(t)$ is the minimal polynomial of $T$ (developed in 01.01.11). Over $K[t]$, which is a principal ideal domain, $m_T$ factors uniquely as

$$m_T(t)=\prod _{i=1}^r{p}_i(t{)}^{e_i},$$

with $p_1,\dots ,p_r$ distinct monic irreducible polynomials and $e_i\ge 1$. The subspaces $\ker p_i(T{)}^{e_i}$ are the primary components of $V$ under $T$. When $K$ contains all eigenvalues, each $p_i(t)=t-{\lambda }_i$ and the primary components are the generalised eigenspaces $\ker (T-{\lambda }_{i}I{)}^{e_i}$.

Notation: $T{|}_W$ is the restriction of $T$ to an invariant subspace $W$; $\overline{T}$ or $T_{V/W}$ is the induced operator on $V/W$; $V=W\oplus U$ denotes an internal direct sum; $E_{\lambda }=\ker (T-\lambda I)$; $G_{\lambda }=\ker (T-\lambda I{)}^e$ for the stabilised index $e$. The symbol $\gcd$ denotes the monic greatest common divisor in $K[t]$, and two polynomials are coprime when their $\gcd$ is $1$.

Counterexamples to common slips

• An invariant subspace need not have an invariant complement. On $K^2$ the operator $T=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)$ keeps the horizontal line $W=\mathrm{span}(e_1)$ to itself, but no second $T$-invariant line exists, since $E_0=\ker T=W$ is the only invariant line. So $T$ admits a block-triangular form but no block-diagonal split.
• The primary decomposition keys off the minimal polynomial, not the characteristic polynomial. Using ${\chi }_T$ in place of $m_T$ gives the same subspaces here because $\ker p_i(T{)}^{e_i}=\ker p_i(T{)}^{a_i}$ once $e_i$ reaches the stabilisation index, but the exponents in $m_T$ are the smallest that produce the full primary component.
• "Each primary component is an eigenspace" fails whenever $e_i>1$. The component $\ker (T-\lambda I{)}^{e_i}$ is the generalised eigenspace; it equals the ordinary eigenspace $\ker (T-\lambda I)$ only when $\lambda$ contributes a square-free factor to $m_T$.

Key theorem with proof Intermediate+

Theorem (primary decomposition; Hoffman-Kunze §6.8 ^{[source pending]}; Shilov Ch. 7 ^{[source pending]}). Let $T:V\to V$ be a linear operator on a finite-dimensional $K$-vector space with minimal polynomial $m_T(t)={\prod }_{i=1}^r{p}_i(t{)}^{e_i}$, the $p_i$ distinct monic irreducibles. Set $W_i=\ker p_i(T{)}^{e_i}$. Then

$$V=W_1\oplus W_2\oplus \cdots \oplus W_r,$$

each $W_i$ is $T$-invariant, the minimal polynomial of $T{|}_{W_i}$ is exactly $p_i(t{)}^{e_i}$, and the projection $P_i:V\to W_i$ along the other summands is a polynomial in $T$.

Proof. For each $i$ put $f_i(t)={\prod }_{j\ne i}{p}_j(t{)}^{e_j}=m_T(t)/p_i(t{)}^{e_i}$. The $r$ polynomials $f_1,\dots ,f_r$ have no common irreducible factor: any irreducible dividing all of them would divide $f_i$ while being coprime to $p_i^{e_i}$, yet $p_i^{e_i}$ is the only factor of $m_T$ absent from $f_i$, so such a common divisor would have to be a unit. Hence $\gcd (f_1,\dots ,f_r)=1$. Because $K[t]$ is a principal ideal domain, Bézout's identity supplies polynomials $g_1,\dots ,g_r\in K[t]$ with

$$\sum _{i=1}^r{g}_i(t)f_i(t)=1.$$

Evaluate at $T$ and define $P_i=g_i(T)f_i(T)$. Then ${\sum }_i{P}_i=I$, and each $P_i$ is a polynomial in $T$, so the $P_i$ commute with $T$ and with one another.

The $P_i$ are projections onto the $W_i$. For $i\ne j$ the product $f_i(t)f_j(t)$ is divisible by $m_T(t)$, because between them $f_i$ and $f_j$ already carry every factor $p_k^{e_k}$ of $m_T$ to at least its full multiplicity (the factor $p_i^{e_i}$ appears in $f_j$, the factor $p_j^{e_j}$ appears in $f_i$, and every other $p_k^{e_k}$ appears in both). Thus $f_i(T)f_j(T)=0$, whence $P_i{P}_j=g_i(T)f_i(T)g_j(T)f_j(T)=0$ for $i\ne j$. Multiplying ${\sum }_k{P}_k=I$ by $P_i$ gives $P_i^2=P_i$, so each $P_i$ is idempotent, and $V={⨁}_i\mathrm{im}{P}_i$ is the internal direct sum attached to a complete system of orthogonal idempotents.

It remains to identify $\mathrm{im}{P}_i$ with $W_i=\ker p_i(T{)}^{e_i}$. If $v\in W_i$, then for $j\ne i$ the polynomial $f_j$ is divisible by $p_i^{e_i}$, so $f_j(T)v=0$ and hence $P_{j}v=0$; from ${\sum }_k{P}_{k}v=v$ this gives $P_{i}v=v$, so $W_i\subseteq \mathrm{im}{P}_i$. Conversely, $p_i(T{)}^{e_i}{P}_i=p_i(T{)}^{e_i}{g}_i(T)f_i(T)$, and $p_i(t{)}^{e_i}{f}_i(t)=m_T(t)$, so $p_i(T{)}^{e_i}{f}_i(T)=m_T(T)=0$; therefore $p_i(T{)}^{e_i}{P}_i=0$, which puts $\mathrm{im}{P}_i\subseteq \ker p_i(T{)}^{e_i}=W_i$. The two inclusions give $\mathrm{im}{P}_i=W_i$, and the direct sum is $V={⨁}_i{W}_i$.

Invariance and the induced minimal polynomial. Each $W_i$ is $T$-invariant because $T$ commutes with $p_i(T{)}^{e_i}$ and so preserves its kernel. Let $q_i(t)$ be the minimal polynomial of $T{|}_{W_i}$. Since $p_i(T{)}^{e_i}$ kills $W_i$, we have $q_i\mid p_i^{e_i}$, so $q_i=p_i^{c_i}$ with $c_i\le e_i$. The polynomial ${\prod }_i{q}_i$ annihilates $T$ on every summand, hence on $V$, so $m_T\mid {\prod }_i{q}_i={\prod }_i{p}_i^{c_i}$. Comparing with $m_T={\prod }_i{p}_i^{e_i}$ forces $c_i\ge e_i$, so $c_i=e_i$ and $q_i=p_i^{e_i}$. $\square$

Bridge. The primary decomposition builds toward the canonical-form theory of 01.01.11: once $V$ splits into the invariant pieces $W_i$ on which $T$ acts with minimal polynomial $p_i^{e_i}$, classifying $T$ reduces to classifying a single operator whose minimal polynomial is a prime power, which is exactly the local problem the Jordan and rational forms solve. This is exactly the linear-algebra face of the Chinese remainder theorem: the coprime factorisation $m_T={\prod }_i{p}_i^{e_i}$ and the Bézout identity ${\sum }_i{g}_i{f}_i=1$ are the ring-level statement $K[t]/(m_T)\cong {\prod }_{i}K[t]/(p_i^{e_i})$, read off on the $K[t]$-module $V$. The foundational reason the projections are polynomials in $T$ is that they are the images of the idempotents of that product ring, so they commute with everything $T$ commutes with and require no choice of basis or inner product. The construction generalises the orthogonal eigenspace splitting of the spectral theorem 01.01.13 by replacing orthogonal projection with algebraic projection, and it appears again in 01.02.05 (the Jordan-Hölder and nilpotent theory) where the same coprime-idempotent mechanism organises a solvable or nilpotent action into its primary blocks. Putting these together, the primary decomposition is the field-agnostic, inner-product-free engine underneath every later canonical form.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib carries invariant submodules, generalised eigenspaces, and the spanning of generalised eigenspaces over an algebraically closed field, but the field-agnostic primary decomposition with polynomial projectors is not packaged as one named theorem. The readable view of the intended statements:

import Mathlib.LinearAlgebra.Eigenspace.Triangularizable
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.FieldTheory.Minpoly.Field

variable {K V : Type*} [Field K] [AddCommGroup V] [Module K V]
  [FiniteDimensional K V] (f : Module.End K V)

/-- A subspace is f-invariant iff it is carried into itself.
Mathlib spelling: `p ≤ p.comap f`. -/
def IsInvariant (p : Submodule K V) : Prop := p ≤ p.comap f

/-- Primary decomposition: if `minpoly K f = ∏ pᵢ ^ eᵢ` with the `pᵢ`
distinct monic irreducibles, then `V = ⨆ i, ker (pᵢ f) ^ eᵢ` as an
internal direct sum, each summand f-invariant, each projection a
polynomial in f. NOT a single named Mathlib theorem. -/
theorem primary_decomposition
    {ι : Type*} [Fintype ι] (p : ι → Polynomial K) (e : ι → ℕ)
    (hirr : ∀ i, Irreducible (p i))
    (hdist : Function.Injective p)
    (hmin : minpoly K f = ∏ i, (p i) ^ (e i)) :
    DirectSum.IsInternal
      (fun i => LinearMap.ker (((p i) ^ (e i)).aeval f)) :=
  sorry  -- Bézout: ∑ gᵢ fᵢ = 1 with fᵢ = m / pᵢ^eᵢ; projectors Pᵢ = gᵢ(f) fᵢ(f)

The proof gap to a clean contribution: (i) form $f_i=m_T/p_i^{e_i}$ and obtain the Bézout identity ${\sum }_i{g}_i{f}_i=1$ from IsCoprime on the pairwise-coprime $p_i^{e_i}$; (ii) define $P_i=(g_i{f}_i)(T)$ and show ${\sum }_i{P}_i=1$, $P_i{P}_j=0$ for $i\ne j$, $P_i^2=P_i$; (iii) identify $\mathrm{im}{P}_i$ with LinearMap.ker ((p i ^ e i).aeval f) to package DirectSum.IsInternal. Each step is reachable from Mathlib's coprimeness and eigenspace machinery, but no current named lemma presents the primary decomposition with its polynomial projectors or derives the Jordan-Chevalley split as a corollary.

Advanced results Master

Theorem (generalised eigenspace decomposition over an algebraically closed field; Shilov Ch. 7 ^{[source pending]}). Let $K$ be algebraically closed and $T$ act on a finite-dimensional $V$ with distinct eigenvalues ${\lambda }_1,\dots ,{\lambda }_r$. Writing $m_i$ for the multiplicity of ${\lambda }_i$ in $m_T$,

$$V=\underset{i=1}{\overset{r}{⨁}}\ker (T-{\lambda }_{i}I{)}^{m_i},$$

and $\dim \ker (T-{\lambda }_{i}I{)}^{m_i}$ equals the algebraic multiplicity of ${\lambda }_i$ in the characteristic polynomial. This is the primary decomposition specialised to linear irreducible factors. The chain $\ker (T-{\lambda }_{i}I)\subseteq \ker (T-{\lambda }_{i}I{)}^2\subseteq \cdots$ stabilises at index $m_i$ — the exponent in the minimal polynomial — and the stable space is the full generalised eigenspace $G_{{\lambda }_i}$. On each $G_{{\lambda }_i}$ the operator $N_i=(T-{\lambda }_{i}I){|}_{G_{{\lambda }_i}}$ is nilpotent, which is the entry point to the Jordan refinement of 01.01.11.

Theorem (Jordan-Chevalley decomposition). Over a perfect field $K$ with $m_T$ splitting into linear factors over $K$, there is a unique pair of operators $S,N$ on $V$ with

$$T=S+N,SN=NS,S\text{ semisimple (diagonalisable)},N\text{ nilpotent},$$

and moreover $S$ and $N$ are polynomials in $T$ with zero constant term in the appropriate sense. Concretely, $S={\sum }_i{\lambda }_i{P}_i$ acts as the scalar ${\lambda }_i$ on the generalised eigenspace $G_{{\lambda }_i}$, with $P_i$ the primary projection, and $N=T-S$ acts as $T-{\lambda }_{i}I$ on $G_{{\lambda }_i}$, hence nilpotent there. The semisimple part $S$ records the eigenvalue data and the nilpotent part $N$ records the failure of diagonalisability. Because $S={\sum }_i{\lambda }_i{P}_i$ is a polynomial in $T$ (each $P_i$ is), the splitting is intrinsic: no basis, no inner product, no diagonalising change of coordinates is chosen.

Theorem (cyclic / rational canonical refinement; Frobenius 1878 ^{[source pending]}; Hoffman-Kunze Ch. 7 ^{[source pending]}). Each primary component $W_i=\ker p_i(T{)}^{e_i}$ decomposes further as a direct sum of $T$-cyclic subspaces $K[t]\cdot v$, each isomorphic as a $K[t]$-module to $K[t]/(p_i^d)$ for some $d\le e_i$, and on a cyclic subspace the matrix of $T$ is the companion matrix of $p_i^d$. This is the structure theorem for the finitely generated torsion $K[t]$-module $V$ over the principal ideal domain $K[t]$. The invariant-factor form collects the cyclic pieces into a chain $d_1(t)\mid d_2(t)\mid \cdots \mid d_s(t)$ with $d_s=m_T$ and ${\prod }_j{d}_j={\chi }_T$; the elementary-divisor form lists the prime powers $p_i^d$ directly. Over an algebraically closed field the elementary divisors are $(t-\lambda {)}^d$ and the companion matrices become Jordan blocks, recovering 01.01.11; over a general field the companion matrices stay, giving the rational canonical form valid without enlarging $K$.

Theorem (module-theoretic reading). The data $(V,T)$ is the same as a finitely generated torsion module over $K[t]$, with $t$ acting as $T$; the primary decomposition is the decomposition of this module into its $p_i$-primary components, and the cyclic refinement is the structure theorem for modules over a PID. The annihilator of the module is $(m_T)$, its order ideal is $({\chi }_T)$, and the equivalence $K[t]/(m_T)\cong {\prod }_{i}K[t]/(p_i^{e_i})$ is the Chinese remainder isomorphism whose idempotents are the projections $P_i$. This identification turns every canonical-form theorem of linear algebra into a special case of the classification of finitely generated modules over a principal ideal domain.

Synthesis. The single operative fact is that $V$ carries a $K[t]$-module structure with $t$ acting as $T$, and the minimal polynomial $m_T={\prod }_i{p}_i^{e_i}$ is the annihilator of that module. The coprimeness of the prime-power factors, expressed through the Bézout identity ${\sum }_i{g}_i{f}_i=1$, manufactures a complete system of commuting idempotents $P_i=(g_i{f}_i)(T)$, and these idempotents are the primary projections; this is the Chinese remainder theorem realised on the module $V$. From that one construction the entire canonical-form hierarchy unfolds: the primary decomposition $V={⨁}_i\ker p_i(T{)}^{e_i}$ is immediate, the Jordan-Chevalley split $T=S+N$ with $S={\sum }_i{\lambda }_i{P}_i$ a polynomial in $T$ is its scalar-versus-nilpotent reading, and the cyclic refinement of each primary component is the PID structure theorem that produces the rational canonical form over any field and the Jordan form over an algebraically closed one. Because the projections are polynomials in $T$, every step is intrinsic and basis-free, which is why the decomposition survives the loss of an inner product that the spectral theorem 01.01.13 required: orthogonal projection is replaced by algebraic projection, diagonalisability is replaced by a nilpotent remainder, and the result holds over fields where eigenvectors may not even exist. The primary decomposition, the generalised eigenspace splitting, the Jordan-Chevalley decomposition, and the cyclic / rational canonical form are four readings of the single statement that a finitely generated torsion $K[t]$-module is the direct sum of its primary components.

Full proof set Master

Proposition (uniqueness of the Jordan-Chevalley decomposition). Let $K$ be a field over which $m_T$ splits into linear factors, and suppose $T=S+N=S^{\prime }+N^{\prime }$ are two decompositions with $S,S^{\prime }$ semisimple, $N,N^{\prime }$ nilpotent, $SN=NS$, and $S^{\prime }{N}^{\prime }=N^{\prime }{S}^{\prime }$. Then $S=S^{\prime }$ and $N=N^{\prime }$.

Proof. From $T=S+N$ with $S,N$ commuting, both $S$ and $N$ commute with $T$; in fact, by the primary decomposition $S={\sum }_i{\lambda }_i{P}_i$ is a polynomial in $T$, so $S$ and $N=T-S$ commute with every operator commuting with $T$. In particular $S$ and $N$ commute with $S^{\prime }$ and $N^{\prime }$, since $S^{\prime }=h(T)$ and $N^{\prime }=T-S^{\prime }$ are themselves polynomials in $T$ by the same construction applied to the second decomposition.

Consider $S-S^{\prime }=N^{\prime }-N$. The left side is a difference of two commuting semisimple operators, hence semisimple: commuting semisimple (diagonalisable) operators are simultaneously diagonalisable, and a difference of simultaneously diagonalisable operators is diagonalisable. The right side is a difference of two commuting nilpotent operators, hence nilpotent: if $N^a=0$, $N^{\prime b}=0$, and $N{N}^{\prime }=N^{\prime }N$, the binomial expansion of $(N^{\prime }-N{)}^{a+b}$ has every term containing $N^a$ or $N^{\prime b}$, so $(N^{\prime }-N{)}^{a+b}=0$. An operator that is both semisimple and nilpotent has minimal polynomial both square-free and a power of $t$, so its minimal polynomial is $t$, forcing the operator to be $0$. Therefore $S-S^{\prime }=0$ and $N-N^{\prime }=0$. $\square$

Proposition (the primary projections are the unique idempotents that are polynomials in $T$ and project onto the components). Let $V={⨁}_i{W}_i$ be the primary decomposition of $T$. The projections $P_i$ onto $W_i$ along ${⨁}_{j\ne i}{W}_j$ are the unique operators that are simultaneously (a) polynomials in $T$, (b) idempotent, (c) satisfy $\mathrm{im}{P}_i=W_i$ and $\ker P_i\supseteq W_j$ for $j\ne i$.

Proof. Existence is the key theorem: $P_i=g_i(T)f_i(T)$ has all three properties. For uniqueness, suppose $Q_i$ also satisfies (a)–(c). Both $P_i$ and $Q_i$ act as the identity on $W_i$ and as $0$ on each $W_j$ with $j\ne i$, since (c) forces this on each summand. Because $V={⨁}_j{W}_j$, an operator is determined by its action on each summand, so $P_i=Q_i$ as operators. The hypothesis that $Q_i$ is a polynomial in $T$ is not even needed for this uniqueness, but it is what makes $P_i$ canonical: the projections attached to any $T$-invariant direct-sum decomposition need not be polynomials in $T$, whereas the primary projections always are, because they descend from the idempotents of $K[t]/(m_T)\cong {\prod }_{i}K[t]/(p_i^{e_i})$. $\square$

Proposition (the induced operator on a quotient has minimal polynomial dividing $m_T$). Let $W$ be $T$-invariant and $\overline{T}$ the induced operator on $V/W$. Then $m_{\overline{T}}\mid m_T$ and $m_{T{|}_W}\mid m_T$, and $m_T=\mathrm{lcm}(m_{T{|}_W},m_{\overline{T}})$ when $V=W\oplus U$ with $U$ also invariant, taking $\overline{T}\cong T{|}_U$.

Proof. For any polynomial $p$, $p(\overline{T})(v+W)=p(T)v+W$. Taking $p=m_T$ gives $p(T)v=0$ for all $v$, so $p(\overline{T})=0$ on $V/W$, whence $m_{\overline{T}}\mid m_T$. Likewise $m_T(T{|}_W)=(m_T(T)){|}_W=0$, so $m_{T{|}_W}\mid m_T$. When $V=W\oplus U$ with both summands invariant, a polynomial $p$ annihilates $T$ iff it annihilates both $T{|}_W$ and $T{|}_U$, because $p(T)=p(T{|}_W)\oplus p(T{|}_U)$ on the block-diagonal form; the smallest such monic $p$ is the least common multiple of $m_{T{|}_W}$ and $m_{T{|}_U}$, and $\overline{T}\cong T{|}_U$ via the isomorphism $V/W\cong U$. Hence $m_T=\mathrm{lcm}(m_{T{|}_W},m_{T{|}_U})$. $\square$

Proposition (Cayley-Hamilton from the primary decomposition). With ${\chi }_T={\prod }_i{p}_i^{a_i}$ and $m_T={\prod }_i{p}_i^{e_i}$ over an algebraically closed field, $e_i\le a_i$ for each $i$, hence $m_T\mid {\chi }_T$ and ${\chi }_T(T)=0$.

Proof. On the generalised eigenspace $G_{{\lambda }_i}=\ker (T-{\lambda }_{i}I{)}^{e_i}$ the restriction $T{|}_{G_{{\lambda }_i}}$ has minimal polynomial $(t-{\lambda }_i{)}^{e_i}$ by the key theorem, and its characteristic polynomial is $(t-{\lambda }_i{)}^{\dim G_{{\lambda }_i}}$ because ${\lambda }_i$ is its only eigenvalue. The index of nilpotency $e_i$ of $(T-{\lambda }_{i}I){|}_{G_{{\lambda }_i}}$ is at most $\dim G_{{\lambda }_i}=a_i$, since a nilpotent operator on a $d$-dimensional space satisfies $N^d=0$. Thus $e_i\le a_i$. Multiplying over $i$, $m_T={\prod }_i(t-{\lambda }_i{)}^{e_i}$ divides ${\chi }_T={\prod }_i(t-{\lambda }_i{)}^{a_i}$, and since $m_T(T)=0$ on each component hence on $V$, also ${\chi }_T(T)=0$. The primary decomposition thereby recovers the Cayley-Hamilton identity of 01.01.08 as a divisibility consequence rather than an independent computation. $\square$

Connections Master

The primary decomposition is the inner-product-free generalisation of the orthogonal eigenspace splitting of the spectral theorem 01.01.13. Where the spectral theorem requires a normal operator and produces orthogonal projections onto eigenspaces, the primary decomposition requires only a minimal polynomial and produces algebraic projections — polynomials in $T$ — onto generalised eigenspaces, dropping both normality and the inner product, and it remains valid over fields with no eigenvectors at all.

The cyclic refinement of each primary component is exactly the input to the Jordan and minimal-polynomial theory of 01.01.11: that unit takes a single operator with prime-power minimal polynomial $(t-\lambda {)}^e$ and builds its Jordan chains, which is the per-component problem the primary decomposition isolates. The rational canonical form named there over a general field is the companion-matrix assembly of the same cyclic pieces.

The coprime-idempotent mechanism reappears in the group setting of 01.02.05 (solvable, nilpotent, and Jordan-Hölder structure): the primary decomposition of a single operator and the decomposition of a nilpotent or solvable linear action into characteristic blocks are two instances of the same use of commuting idempotents and stable filtrations to break a hard action into local pieces.

Historical & philosophical context Master

The classification of a single linear substitution by its canonical block form is due to Camille Jordan, whose 1870 Traité des substitutions et des équations algébriques gave the canonical form of a linear substitution first over finite fields, in the service of Galois theory, and then over the complex numbers ^{[Jordan 1870]}. The decomposition that underlies the block form — the splitting of the space according to the prime-power factors of an annihilating polynomial — was placed on its modern algebraic footing by Frobenius, whose 1878 memoir on linear substitutions and bilinear forms introduced the minimal polynomial, the invariant factors, and the rational canonical form, valid over an arbitrary field and not requiring the eigenvalues to lie in it ^{[Frobenius 1878]}. The recognition that all of this is the structure theory of finitely generated modules over the principal ideal domain $K[t]$, with the primary decomposition as the coprime-primary splitting and the canonical forms as the cyclic decomposition, is the ideal-theoretic viewpoint codified by Krull and the abstract-algebra tradition of the 1920s and 1930s ^{[Krull 1935]}. Shilov's Linear Algebra presents the invariant-subspace and canonical-form development in the concrete operator language used here, while Hoffman and Kunze give the primary decomposition theorem in the coprime-annihilator form whose proof this unit follows.

Bibliography Master

@book{Jordan1870,
  author    = {Jordan, Camille},
  title     = {Trait{\'e} des substitutions et des {\'e}quations alg{\'e}briques},
  publisher = {Gauthier-Villars},
  address   = {Paris},
  year      = {1870}
}

@article{Frobenius1878,
  author  = {Frobenius, Ferdinand Georg},
  title   = {{\"U}ber lineare Substitutionen und bilineare Formen},
  journal = {Journal f{\"u}r die reine und angewandte Mathematik},
  volume  = {84},
  year    = {1878},
  pages   = {1--63}
}

@book{Krull1935,
  author    = {Krull, Wolfgang},
  title     = {Idealtheorie},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  publisher = {Springer},
  address   = {Berlin},
  year      = {1935}
}

@book{HoffmanKunze1971,
  author    = {Hoffman, Kenneth and Kunze, Ray},
  title     = {Linear Algebra},
  edition   = {2nd},
  publisher = {Prentice-Hall},
  address   = {Englewood Cliffs, NJ},
  year      = {1971}
}

@book{Shilov1977,
  author    = {Shilov, Georgi E.},
  title     = {Linear Algebra},
  publisher = {Dover Publications},
  address   = {New York},
  year      = {1977},
  note      = {Translation of the 1971 Russian edition, transl. R. A. Silverman}
}

@book{Jacobson1985,
  author    = {Jacobson, Nathan},
  title     = {Basic Algebra I},
  edition   = {2nd},
  publisher = {W. H. Freeman},
  address   = {New York},
  year      = {1985}
}

@book{Lang2002,
  author    = {Lang, Serge},
  title     = {Algebra},
  edition   = {3rd, revised},
  series    = {Graduate Texts in Mathematics},
  volume    = {211},
  publisher = {Springer},
  year      = {2002}
}