Read-back: HefferonLinAlg.jordanBlock_count_from_kernel_dims

What the statement asserts

Fix two natural numbers $n$ and $k$ (both implicit, both allowed to be $0$), a square complex matrix

$$A \in M_{n\times n}(\mathbb{C})$$

indexed by $\{0,1,\dots,n-1\}$, a function $\mathrm{sz} : \{0,\dots,k-1\} \to \mathbb{N}$ assigning a natural number size to each of $k$ indices, and a function $\lambda : \{0,\dots,k-1\} \to \mathbb{C}$ assigning a complex number to each of those same $k$ indices. Assume the hypothesis $h$, spelled out below, that $(\mathrm{sz}, \lambda)$ is a Jordan form of $A$. Then the claim is: for every complex number $\mu$ and every natural number $r$,

$$\#\bigl\{\, i \in \{0,\dots,k-1\} \;:\; \mathrm{sz}(i) = r+1 \ \text{ and } \ \lambda(i) = \mu \,\bigr\} \;+\; d(\mu, r) \;+\; d(\mu, r+2) \;=\; 2\, d(\mu, r+1),$$

where $d(\mu, s)$ abbreviates the quantity $\operatorname{kerDim}(A,\mu,s)$ defined below. Both $\mu$ and $r$ are quantified inside the statement (they are explicit arguments), so a single instance of the theorem covers all of them; $n$, $k$, $A$, $\mathrm{sz}$, $\lambda$ are quantified outside as implicit arguments.

Unfolding every notion the statement depends on

The kernel dimension $d(\mu,s) = \operatorname{kerDim}(A,\mu,s)$

For a matrix $A$ over a field $K$ (here $K = \mathbb{C}$), a scalar $\mu$, and a natural number $s$,

$$d(\mu,s) \;=\; \dim_{\mathbb{C}} \ker\Bigl( x \mapsto (A - \mu I)^{s}\, x \Bigr),$$

i.e. the $\mathbb{C}$-dimension (as a natural number, via Module.finrank) of the kernel of the linear map on the coordinate space $\mathbb{C}^{n}$ given by left multiplication by the matrix power $(A - \mu I)^{s}$. Here $\mu I$ is the scalar $\mu$ acting on the identity matrix, and the power is matrix power with the standard convention $(A-\mu I)^{0} = I$; consequently $d(\mu, 0) = \dim \ker(\mathrm{id}) = 0$ for every $\mu$.

The hypothesis $h$: "$(\mathrm{sz},\lambda)$ is a Jordan form of $A$"

This is the conjunction of two conditions.

1. Every block is nonempty: $\mathrm{sz}(i) > 0$ for all $i \in \{0,\dots,k-1\}$.

2. There exists a re-indexing bijection

$$e : \{0,\dots,n-1\} \;\xrightarrow{\ \sim\ }\; \coprod_{i=0}^{k-1} \{0,\dots,\mathrm{sz}(i)-1\}$$

(a bijection between the index set of $A$ and the disjoint union of the block index sets — its mere existence forces $n = \sum_{i} \mathrm{sz}(i)$), such that $A$ is similar to the matrix $J^{e}$ obtained by transporting the block-diagonal Jordan matrix $J$ along $e$, i.e. the $n \times n$ matrix with entries $J^{e}_{a b} = J_{\,e(a)\, e(b)}$.

Here:

• The Jordan matrix $J = \operatorname{jordanMatrix}(\mathrm{sz},\lambda)$ is the block-diagonal matrix, indexed by pairs $(i, a)$ with $i \in \{0,\dots,k-1\}$ and $a \in \{0,\dots,\mathrm{sz}(i)-1\}$, whose $i$-th diagonal block is the $\mathrm{sz}(i) \times \mathrm{sz}(i)$ Jordan block with eigenvalue $\lambda(i)$, and whose off-diagonal blocks are zero: $J_{(i,a),(j,b)} = 0$ whenever $i \neq j$.

• The Jordan block $\operatorname{jordanBlock}(m, c)$ is the $m \times m$ matrix whose $(p,q)$ entry is

$$\begin{cases} c & \text{if } p = q,\\ 1 & \text{if } p = q+1,\\ 0 & \text{otherwise,} \end{cases}$$

so the ones sit on the subdiagonal (below the diagonal), not the superdiagonal.

• Similarity of two square matrices $X, Y$ over a commutative ring means: there exists a matrix $P$ of the same shape whose determinant is a unit (over $\mathbb{C}$: $\det P \neq 0$) with

$$P^{-1} X P = Y,$$

where $P^{-1}$ is the standard total inverse operation (well-behaved here because $\det P$ is assumed invertible). Note the direction: the hypothesis says $P^{-1} A P = J^{e}$, i.e. $A$ is conjugated into the Jordan matrix.

Arithmetic and typing notes

• The counted quantity $\#\{\, i : \mathrm{sz}(i) = r+1 \text{ and } \lambda(i) = \mu \,\}$ is the cardinality of a subset of the finite index set $\{0,\dots,k-1\}$ — a count of block indices, not of blocks weighted by size. The condition is an exact equality $\mathrm{sz}(i) = r+1$ of natural numbers together with an exact equality $\lambda(i) = \mu$ of complex numbers.
• Every quantity in the displayed equation is a natural number, and the equation is an equality in $\mathbb{N}$: the block count, the three kernel dimensions, the two additions, and the multiplication by $2$. No subtraction occurs anywhere in the statement, so no truncated-natural-subtraction behaviour is in play. The equation is written in the "all terms on the correct side" form

$$\text{count} + d(\mu,r) + d(\mu,r+2) = 2\,d(\mu,r+1)$$

rather than as a difference.

• The indices $r$, $r+1$, $r+2$ are all natural numbers; the smallest instance is $r = 0$, which reads

$$\#\{\, i : \mathrm{sz}(i) = 1 \text{ and } \lambda(i) = \mu \,\} + 0 + d(\mu,2) = 2\,d(\mu,1),$$

since $d(\mu,0) = 0$ as noted above.

• $\mu$ ranges over all of $\mathbb{C}$; nothing requires $\mu$ to be an eigenvalue of $A$ or to appear among the values $\lambda(i)$. For such a $\mu$ all three kernel dimensions and the count are $0$.

Degenerate cases silently included

• $k = 0$: the index set of blocks is empty, so the nonemptiness condition $\mathrm{sz}(i) > 0$ holds vacuously, the disjoint union is empty, and the existence of the bijection $e$ forces $n = 0$. Then $A$ is the empty $0 \times 0$ matrix, the count is $0$, all kernel dimensions are $0$, and the equation reads $0 + 0 + 0 = 2\cdot 0$.
• $n = 0$: the underlying space is the zero space, so every kernel dimension is $0$.
• The hypothesis $h$ is not satisfiable for arbitrary combinations of the implicit data — for instance, whenever $n \neq \sum_i \mathrm{sz}(i)$, or whenever $A$ is not similar to the indicated block-diagonal matrix, no such $e$ and $P$ exist, and the theorem says nothing about those instances.
• The conclusion is stated about kernel dimensions of $A$ itself, not of the Jordan matrix $J$; the link between them is carried entirely by the hypothesis $h$.
• Nothing in the statement requires the eigenvalues $\lambda(i)$ to be distinct across blocks, nor the sizes $\mathrm{sz}(i)$ to be sorted or distinct; repeated $(\mathrm{sz}(i), \lambda(i))$ pairs are allowed and each contributes separately to the count.

Proof status

The declaration's proof body is sorry: no proof is supplied.