Kernel Dimensions of a Jordan Block

The Action of a Nilpotent Jordan Block

For a nilpotent Jordan block, taking higher powers eventually gives the zero matrix. Before that happens, each power moves the nonzero entries farther above the main diagonal, and its kernel grows. We will derive the exact dimension of the kernel for every positive power and explain how the pattern identifies the length of the block.

Let \(k\) be a positive integer, and work over \(\mathbb{R}\). We use the convention that the nilpotent Jordan block of size \(k\) has ones on the first superdiagonal and zeros everywhere else. The standard basis of \(\mathbb{R}^k\) is \(\mathbf{e}_1,\dots,\mathbf{e}_k\). The kernel of a matrix \(\mathbf{A}\) is the set of vectors \(\mathbf{x}\) satisfying \(\mathbf{A}\mathbf{x}=\mathbf{0}\); its dimension is also called its nullity.

In particular, \(\mathbf{J}_k(0)\mathbf{e}_1=\mathbf{0}\), while \(\mathbf{J}_k(0)\mathbf{e}_j=\mathbf{e}_{j-1}\) for \(2\le j\le k\). Each application of the block shifts a basis vector down by one index, until the vector reaches \(\mathbf{e}_1\) and is sent to zero. A power applies this shift repeatedly.

The Kernel-Dimension Formula

Proof. Repeatedly applying the basis-vector action gives \(\mathbf{J}_k(0)^r\mathbf{e}_j=\mathbf{0}\) when \(j\le r\), because the shift reaches \(\mathbf{e}_1\) and then gives zero. When \(j>r\), it gives \(\mathbf{J}_k(0)^r\mathbf{e}_j=\mathbf{e}_{j-r}\). For an arbitrary vector \(\mathbf{x}=\sum_{j=1}^{k}x_j\mathbf{e}_j\), linearity therefore gives \(\mathbf{J}_k(0)^r\mathbf{x}=\sum_{j=r+1}^{k}x_j\mathbf{e}_{j-r}\), where the sum is zero if \(r\ge k\). The vectors \(\mathbf{e}_{j-r}\) in this sum have distinct indices, so their linear combination is zero exactly when each coefficient \(x_j\) with \(j>r\) is zero. Thus the kernel is \(\operatorname{span}\{\mathbf{e}_1,\dots,\mathbf{e}_{\min(r,k)}\}\). These are \(\min(r,k)\) distinct standard basis vectors, so they are linearly independent and form a basis of the kernel. Its dimension is consequently \(\min(r,k)\). \(\square\)

The proof also describes the action of each power. For \(r<k\), the \(r\)th power has ones exactly \(r\) places above the main diagonal: it sends \(\mathbf{e}_j\) to \(\mathbf{e}_{j-r}\) when \(j>r\), and sends the other standard basis vectors to zero. When \(r\ge k\), it sends every standard basis vector to zero, so the whole matrix is zero. This makes the kernel formula visible in the entries: the first \(r\) coordinate directions are killed, unless \(r\) has already reached \(k\).

How the Sequence Records the Block Length

For one block, write \(d_r=\dim\ker\bigl(\mathbf{J}_k(0)^r\bigr)\) for each positive integer \(r\). The theorem says \(d_r=\min(r,k)\). So the sequence starts \(1,2,3,\dots\), increasing by one at each step, until it reaches \(k\); after that it stays equal to \(k\). In particular, \(d_r=k\) exactly when \(r\ge k\). The first positive integer \(r\) for which the kernel dimension is \(k\) is therefore the block length.

Proof. By the theorem, \(d_r=\min(r,k)\). If \(r<k\), then \(d_r=r<k\), so the dimension has not yet reached \(k\). If \(r\ge k\), then \(d_r=k\). Thus the smallest positive \(r\) with \(d_r=k\) is precisely \(r=k\). \(\square\)

This conclusion concerns a single block: the sequence records the number of basis directions that have been killed after each number of shifts, and it reaches the full space dimension when every direction is killed. The time it takes to reach that point is the block length. The same sequence also expresses the nilpotency index: the smallest positive exponent giving the zero matrix is \(k\). Indeed, for \(r<k\), the vector \(\mathbf{e}_{r+1}\) is sent to \(\mathbf{e}_1\ne\mathbf{0}\), whereas for \(r\ge k\), every basis vector is sent to zero.

Common Pitfall: Counting Diagonal Offsets

Do not confuse the power number with the number of places above the main diagonal. The first power has ones one place above the main diagonal; the second power has ones two places above it. For example, in the size-five block, the third power has nonzero entries in positions \((1,4)\) and \((2,5)\), three columns to the right of the main diagonal positions in those rows. It does not have ones on the first or second superdiagonal.

A related error is to conclude that the kernel has dimension \(r\) for every positive \(r\), without considering the size \(k\). For a block of size \(3\), that would incorrectly assign dimension \(4\) to the kernel of the fourth power. But \(\mathbf{J}_3(0)^3=\mathbf{O}\), so its fourth power is also zero and its kernel is all of \(\mathbb{R}^3\), which has dimension \(3\). The minimum in \(\min(r,k)\) accounts for this plateau and ensures the dimension never exceeds the size of the space.

When using the formula, first identify the block size \(k\) and the power \(r\), then compare them. If \(r<k\), the dimension is \(r\); if \(r\ge k\), the dimension is \(k\). Keeping the cases explicit prevents an increasing sequence from being extended beyond the dimension of the underlying space.