What Kernel Dimensions Record
A Jordan form displays the structure of a matrix in blocks, but the order of those blocks is not important: permuting them changes their arrangement, not their sizes. The question here is whether two Jordan forms of the same matrix could have genuinely different block sizes for the same eigenvalue. The answer is no. The dimensions of the kernels of successive powers of \(\mathbf{A}-\lambda\mathbf{I}\) encode exactly how many blocks there are of each size.
We work over \(\mathbb{C}\), with \(\mathbf{A}\in\mathbb{C}^{n\times n}\) and \(\lambda\) an eigenvalue of \(\mathbf{A}\). The Jordan Form Existence Theorem says that \(\mathbf{A}\) is similar to a block diagonal matrix whose blocks are Jordan blocks. A Jordan block of size \(s\) for \(\lambda\), denoted \(\mathbf{J}_s(\lambda)\), has \(\lambda\) on its diagonal, \(1\) directly above the diagonal, and \(0\) elsewhere. Define \(d_0=0\), and, for \(k\geq 1\), define
The sequence \(d_0,d_1,d_2,\ldots\) is nondecreasing and eventually constant. Its increments have a simple interpretation: each Jordan block for \(\lambda\) contributes one to the increment for every power up to its size, then contributes nothing further. Thus the first differences count blocks of size at least \(k\); subtracting consecutive such counts gives the number of blocks of size exactly \(k\).
Why a Single Block Contributes This Way
We first calculate the kernel dimensions for one block. Subtracting \(\lambda\mathbf{I}\) from \(\mathbf{J}_s(\lambda)\) leaves a matrix \(\mathbf{N}\) with \(1\) directly above the diagonal and \(0\) elsewhere. With the standard coordinate vectors \(\mathbf{e}_1,\ldots,\mathbf{e}_s\), it acts by \(\mathbf{N}\mathbf{e}_1=\mathbf{0}\) and \(\mathbf{N}\mathbf{e}_j=\mathbf{e}_{j-1}\) for \(2\leq j\leq s\). Consequently, \(\mathbf{N}^k\mathbf{e}_j=\mathbf{0}\) when \(j\leq k\), and \(\mathbf{N}^k\mathbf{e}_j=\mathbf{e}_{j-k}\) when \(j>k\). The first \(\min(k,s)\) coordinate vectors therefore form a basis for the kernel.
For a block belonging to a different eigenvalue \(\mu\ne\lambda\), the corresponding block of \(\mathbf{J}_s(\mu)-\lambda\mathbf{I}\) is invertible. Indeed, it has the form \(c\mathbf{I}+\mathbf{N}\), where \(c=\mu-\lambda\ne0\) and \(\mathbf{N}^s=\mathbf{O}\); its inverse is \(c^{-1}\sum_{j=0}^{s-1}(-\mathbf{N}/c)^j\). Its powers have zero kernel, so such a block contributes nothing to \(d_k\). Since the kernel of a block diagonal matrix is the direct sum of the kernels of its blocks, only the blocks for \(\lambda\) contribute to \(d_k\).
Worked Example: Reading the Counts from Three Blocks
Suppose the Jordan blocks for \(\lambda\) have sizes \(1,1,3\). For each \(k\), add the contributions \(\min(k,s)\) from the three blocks:
The increments are \(c_1=3\), \(c_2=1\), \(c_3=1\), and \(c_4=0\). Thus three blocks have size at least \(1\), one has size at least \(2\), and one has size at least \(3\). Taking differences of these counts gives \(b_1=3-1=2\), \(b_2=1-1=0\), and \(b_3=1-0=1\). So the recovered sizes are \(1,1,3\).
Check: The reconstructed blocks give \(d_1=1+1+1=3\), \(d_2=1+1+2=4\), and \(d_3=1+1+3=5\), matching the sequence calculated from the original blocks.
The Uniqueness Theorem
Proof. Consider a Jordan form of \(\mathbf{A}\). For a block of size \(s\) belonging to \(\lambda\), the block calculation shows that its contribution to \(d_k\) is \(\min(k,s)\). The difference between its contributions at \(k\) and \(k-1\) is \(1\) if \(s\geq k\), and \(0\) if \(s<k\). Blocks for other eigenvalues contribute \(0\), as shown above. Adding the contributions of all blocks proves that \(c_k=d_k-d_{k-1}\) counts exactly the blocks of size at least \(k\). A block has size exactly \(k\) precisely when it is counted among blocks of size at least \(k\), but not among blocks of size at least \(k+1\). Therefore the number of size-\(k\) blocks is \(b_k=c_k-c_{k+1}\). This gives \(b_k=2d_k-d_{k-1}-d_{k+1}\) for every \(k\). The formula depends only on the kernel dimensions of \(\mathbf{A}\), not on the chosen Jordan form. Applying it to any two Jordan forms gives the same number of blocks of every size, which proves uniqueness up to order. \(\square\)
The proof also explains why the sequence eventually stops changing. There are finitely many Jordan blocks, so there is a largest size \(L\). For \(k>L\), every block contributes its full size, giving \(d_k=d_L\) and \(c_k=0\). In particular, setting \(c_{L+1}=0\) makes the exact-size formula apply to the largest blocks as well.
Worked Example: Recovering Four Different Sizes
Suppose the kernel-dimension data for an eigenvalue are \(d_0=0\), \(d_1=4\), \(d_2=7\), \(d_3=9\), \(d_4=10\), and \(d_5=10\). First calculate the increments:
The number of blocks of each exact size is then \(b_1=4-3=1\), \(b_2=3-2=1\), \(b_3=2-1=1\), and \(b_4=1-0=1\). Since \(c_5=0\), there are no blocks of size \(5\) or larger. The block sizes are therefore \(1,2,3,4\).
Check: These four blocks give \(d_1=1+1+1+1=4\), \(d_2=1+2+2+2=7\), \(d_3=1+2+3+3=9\), and \(d_4=1+2+3+4=10\). At \(k=5\), each block still contributes its full size, so \(d_5=10\). Every value agrees with the given data.
Similarity Preserves the Kernel-Dimension Sequence
The theorem concerns Jordan forms of the same matrix, so it matters that changing coordinates does not change these kernel dimensions. If \(\mathbf{B}=\mathbf{P}^{-1}\mathbf{A}\mathbf{P}\), where \(\mathbf{P}\) is invertible, then
For a vector \(\mathbf{x}\), the equation \((\mathbf{B}-\lambda\mathbf{I})^k\mathbf{x}=\mathbf{0}\) holds exactly when \((\mathbf{A}-\lambda\mathbf{I})^k\mathbf{P}\mathbf{x}=\mathbf{0}\). The invertible map \(\mathbf{P}\) therefore carries the first kernel bijectively onto the second. Their dimensions agree.
Worked Example: Kernel Data Distinguish Two Proposed Forms
Compare two proposed lists of block sizes for the same eigenvalue: \(2,2,3\) and \(1,3,3\). Each list has total size \(7\), and each has three blocks, so the total dimension and \(d_1\) agree. Calculate the next dimensions by adding \(\min(k,s)\) for each block.
The two lists give different values of \(d_2\), so they cannot both describe Jordan forms of the same matrix for this eigenvalue. This illustrates why the total dimension and the number of blocks alone are insufficient; the intermediate kernel dimensions provide further information.
Check: For the first list the increments are \(3,3,1,0\), yielding \(b_1=0\), \(b_2=2\), and \(b_3=1\). For the second list they are \(3,2,2,0\), yielding \(b_1=1\), \(b_2=0\), and \(b_3=2\). Each calculation recovers its original list and confirms that the two lists have different kernel-dimension data.
Common Mistake: Confusing “At Least” with “Exactly”
The increments \(c_k=d_k-d_{k-1}\) do not count blocks of size exactly \(k\). They count blocks of size at least \(k\). For the data \(d_0=0,d_1=3,d_2=4,d_3=5,d_4=5\), the increments are \(3,1,1,0\). Treating those numbers as exact-size counts would incorrectly claim three blocks of size \(1\), one of size \(2\), and one of size \(3\). That claim is inconsistent: it gives five blocks and a total size of \(3+2+3=8\), even though the stabilized kernel dimension is \(5\).
The correction is to take differences of the increments: \(b_1=3-1=2\), \(b_2=1-1=0\), and \(b_3=1-0=1\). These are two blocks of size \(1\), no blocks of size \(2\), and one block of size \(3\). Their total size is \(1+1+3=5\), consistent with the stabilized kernel dimension. Keeping the two stages distinct—first differences count sizes at least \(k\), second differences recover sizes exactly \(k\)—prevents this error.
Why the Uniqueness Result Matters
A Jordan form is not unique as an ordered block matrix: its blocks can be permuted. What is unique is the collection of block sizes attached to each eigenvalue. The kernel-dimension sequence makes that claim precise and gives a direct way to recover the collection. Each additional power of \(\mathbf{A}-\lambda\mathbf{I}\) reveals how many blocks extend at least one step further in size. The drop between successive extension counts records the blocks that end at that size.
As a result, kernel dimensions can confirm or rule out proposed Jordan structures without relying on the order in which blocks are written. The exact-size formula also provides an internal consistency check: the values \(b_k\) must be nonnegative integers and must eventually be zero. If calculated data violate either condition, they cannot be the kernel dimensions of powers for a matrix with a Jordan form of the stated kind.
Check Your Understanding
Use the block contribution \(\min(k,s)\) and distinguish carefully between blocks of size at least \(k\) and blocks of size exactly \(k\).
- If \(d_0=0,d_1=4,d_2=6,d_3=7,d_4=7\), how many blocks of each size occur?
- For blocks of sizes \(1,2,4\), calculate \(d_1,d_2,d_3,d_4\) and verify that the exact-size formula recovers the list.
- Explain why \(d_k-d_{k-1}\) counts blocks of size at least \(k\), rather than blocks of size exactly \(k\).
- Two proposed block lists have the same number of blocks and the same total size. Must they give the same kernel-dimension sequence? Explain using a block contribution.
- Why do Jordan blocks belonging to an eigenvalue \(\mu\ne\lambda\) contribute zero to \(\dim\ker(\mathbf{A}-\lambda\mathbf{I})^k\)?
