Constructing Jordan Chains for Nilpotent Maps

From Kernel Spaces to Chain Tops

A nilpotent operator eventually sends every vector to zero. The vectors it kills after one application, two applications, or more reveal where chains can begin and how long they can be. The key construction is to choose a top vector of length \(r\) from \(\ker N^r\), while excluding vectors that already belong to a shorter kernel or that have arrived as images from a longer chain. Applying \(N\) repeatedly to a chosen top then produces its chain.

Let \(V\) be a finite-dimensional vector space over \(\mathbb{R}\), and let \(N:V\to V\) be a nilpotent linear map. Nilpotent means that \(N^s=\mathbf{0}\) for some positive integer \(s\), where \(\mathbf{0}\) here denotes the zero map. Take \(s\) to be the least such integer, called the nilpotency index. For each integer \(r\geq 0\), define \(K_r=\ker N^r\), with \(N^0=\operatorname{id}_V\). Thus \(K_0=\{\mathbf{0}\}\), and \(K_s=V\). Also \(K_{s+1}=K_s\), since both equal \(V\).

Being in \(K_r\) is necessary for a top of length \(r\), but it is not enough to identify the new tops of that length. For instance, if \(\mathbf{v}\) is the top of a chain of length \(r+1\), then \(N\mathbf{v}\) lies in \(K_r\) and has length \(r\). It is already part of the longer chain, so it should not be chosen as the top of an additional chain. We need a systematic way to exclude such images.

The Quotient That Selects New Tops

For each \(1\leq r\leq s\), consider the subspace \(K_{r-1}+N(K_{r+1})\) inside \(K_r\). The first term excludes vectors whose chains have length less than \(r\). The second excludes vectors that are images of vectors in \(K_{r+1}\), including the vectors already inside longer chains.

Proof. First, \(N(K_{r+1})\subseteq K_r\), since \(N^{r+1}\mathbf{x}=\mathbf{0}\) implies \(N^r(N\mathbf{x})=\mathbf{0}\). Also \(K_{r-1}\subseteq K_r\), so the stated quotient is defined. The kernel of \(N\) restricted to \(K_j\) is \(K_1\) for every \(j\geq1\). By the Rank-Nullity Theorem, \(\dim N(K_j)=k_j-k_1\). Next, \(K_{r-1}\cap N(K_{r+1})=N(K_r)\): if \(\mathbf{y}=N\mathbf{x}\) lies in that intersection, then \(N^{r-1}\mathbf{y}=\mathbf{0}\), so \(N^r\mathbf{x}=\mathbf{0}\) and \(\mathbf{x}\in K_r\); conversely, \(\mathbf{x}\in K_r\) implies \(N\mathbf{x}\in K_{r-1}\) and \(N\mathbf{x}\in N(K_{r+1})\). Thus the intersection has dimension \(k_r-k_1\). The dimension formula for a sum of two subspaces gives \(\dim(K_{r-1}+N(K_{r+1}))=k_{r-1}+(k_{r+1}-k_1)-(k_r-k_1)=k_{r-1}+k_{r+1}-k_r\). Subtracting this from \(k_r\) gives the asserted quotient dimension. Finally, a nonzero quotient class has a representative \(\mathbf{v}\notin K_{r-1}\), because \(K_{r-1}\) is contained in the denominator. Since \(\mathbf{v}\in K_r\), we have \(N^r\mathbf{v}=\mathbf{0}\), and since \(\mathbf{v}\notin K_{r-1}\), we have \(N^{r-1}\mathbf{v}\ne\mathbf{0}\). Hence \(\mathbf{v}\) is a top vector of length exactly \(r\). \(\square\)

The dimension formula used in the proof follows from rank-nullity: the map from \(U\oplus W\) to \(U+W\) sending \((\mathbf{u},\mathbf{w})\) to \(\mathbf{u}+\mathbf{w}\) has kernel consisting of pairs \((\mathbf{z},-\mathbf{z})\) with \(\mathbf{z}\in U\cap W\). The quotient dimension therefore counts how many new tops of length \(r\) can be selected after the shorter vectors and images from longer kernels are excluded. This is the kernel-growth count for chains of exact length \(r\).

Worked Construction in Four Dimensions

Applying Powers to Produce a Chain

Once a top vector has been chosen, the chain is computed in the order \(N^{r-1}\mathbf{v},\dots,N\mathbf{v},\mathbf{v}\). The top itself is the vector killed after \(r\) applications; the first vector in the written chain is the nonzero vector killed after one application. The next example illustrates each power explicitly.

A Second Kernel-Quotient Example

Common Mistake: Reusing a Vector from a Longer Chain

In the four-dimensional example, \(\mathbf{e}_2\in K_2\setminus K_1\), so it has exact length \(2\). A tempting choice is to select it as a new top for a length-two chain, in addition to selecting \(\mathbf{e}_3\) as a length-three top. That would produce \(\mathbf{e}_1,\mathbf{e}_2\) from \(\mathbf{e}_2\), even though both vectors already occur in the chain \(\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3\) generated by \(\mathbf{e}_3\). The error is treating every vector of exact length \(2\) as a new top. The correction is to exclude \(N(K_3)\): since \(\mathbf{e}_2=N\mathbf{e}_3\), it lies in \(K_1+N(K_3)\), and contributes no new quotient direction.

The quotient construction specifies how many tops to choose at each length and how to avoid this particular duplication. It does not, by itself, establish that all vectors in all selected chains jointly form a basis of \(V\); that requires a further spanning and independence argument. Its immediate practical use is precise: compute the kernels, form the quotient for each length, choose representatives, and apply successive powers to write the chains. The dimensions of those quotients are nonnegative because they are dimensions of actual quotient spaces.