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Let $K$ be a field, $p \geq 1$ , and $A \in M_{p} (K)$ . If $χ_{A} (t) = \prod_{i < p} (t - λ_{i})$ in $K [t]$ , then the transfer-matrix trace series is $\sum_{i < p} (1 - λ_{i} x)^{- 1}$

Statement

Let $K$ be a field, let $p \geq 1$ , and let $A \in M_{p} (K)$ . Suppose that its characteristic polynomial is displayed as a product of linear factors in the base field,

$χ_{A} (t) = \prod_{i < p} (t - λ_{i}) in K [t],$

where roots are repeated according to algebraic multiplicity. Then

$tr (A^{n}) = \sum_{i < p} λ_{i}^{n} (n \geq 0),$

and, formally,

$\sum_{n \geq 0} tr (A^{n}) x^{n} = \sum_{i < p} \frac{1}{1 - λ_{i} x} .$

When $A$ is a transfer matrix, this is the eigenvalue form of its closed-walk trace series.

Facts & Assumptions

Given: A field $K$ , a positive size $p$ , a matrix $A \in M_{p} (K)$ , and the displayed split factorisation of $χ_{A}$ in $K [t]$ .

[L1]

For a transfer matrix, the closed-walk series is $\sum_{n \geq 0} {tr}_{K} (A^{n}) x^{n}$ (Closed walks have trace and logarithmic-derivative generating functions).

[L2]

The matrix $A$ defines the coordinate endomorphism $T_{A} (x) = A x$ (For $A \in M_{n} (R)$ , the coordinate endomorphism $T_{A} : R^{n} \to R^{n}$ , with $\det (T_{A}) : = \det (A)$ and $adj (T_{A}) : = T_{adj (A)}$ ).

[L3]

The characteristic polynomial of an endomorphism equals that of any representing matrix (The basis-independent characteristic polynomial $χ_{T}$ of an endomorphism of a finite-dimensional space, including $χ_{T} = 1$ in dimension zero).

[L4]

The trace of an endomorphism equals the trace of any representing matrix (The basis-independent trace of an endomorphism of a finite-dimensional vector space).

[L5]

If $χ_{T} (t) = \prod_{i < p} (t - λ_{i})$ , then $χ_{q (T)} (y) = \prod_{i < p} (y - q (λ_{i}))$ for every polynomial $q$ (If $χ_{T} (x) = \prod_{i < n} (x - λ_{i})$ in $F [x]$ , then $χ_{p (T)} (y) = \prod_{i < n} (y - p (λ_{i}))$ for every $p \in F [x]$ : the eigenvalues of $p (T)$ are $p (λ_{i})$ , counted with algebraic multiplicity).

[L6]

When the characteristic polynomial splits, the trace is the sum of its roots with algebraic multiplicity (If $χ_{T} (x) = \prod_{i < n} (x - λ_{i})$ in $F [x]$ , then $tr (T) = \sum_{i < n} λ_{i}$ : trace is the sum of the eigenvalues counted with algebraic multiplicity).

[L7]

The case $j = 1$ of the repeated-pole formula is $(1 - λ x)^{- 1} = \sum_{n \geq 0} λ^{n} x^{n}$ (Repeated poles expand formally as $(1 - λ x)^{- j} = \sum_{n \geq 0} (\binom{n + j - 1}{j - 1}) λ^{n} x^{n}$ ).

[L8]

Over a field, the commutative-ring matrix trace equals the published matrix trace (For matrices over a field, the commutative-ring trace agrees with the published matrix trace).

Proof

technique · direct

1.1

Let $T_{A}$ be the coordinate endomorphism from [L2]. By [L3], the given factorisation is also $χ_{T_{A}} (t) = \prod_{i < p} (t - λ_{i})$ .

givenL2L3

2.1

Apply [L5] to $q (t) = t^{n}$ . The roots of $χ_{T_{A}^{n}}$ are $λ_{i}^{n}$ with the displayed multiplicities, so [L6] gives $tr (T_{A}^{n}) = \sum_{i < p} λ_{i}^{n}$ .

step 1.1L5L6

3.1

By [L4] and [L8], the left side in step 2.1 is $tr (A^{n})$ in either trace convention. This proves the coefficient formula, including $n = 0$ .

step 2.1L4L8

4.1

Multiply the coefficient formula by $x^{n}$ , sum formally over $n \geq 0$ , and apply [L7] to each of the finitely many roots; this yields the displayed rational-series identity.

step 3.1L7algebra

5.1

If $A$ is a transfer matrix, [L1] identifies the left side of step 4.1 with its closed-walk trace series. The split factorisation was assumed at the outset and is not inferred from algebraic closure or diagonalisation.

step 4.1L1∎

Depends on

Used by

Closed walks of length n in Kₚ are counted by (p-1)ⁿ+(p-1)(-1)ⁿ Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 115 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

R. P. Stanley, Enumerative Combinatorics, vol. 1, 2nd ed., Corollaries 4.7.3-4.7.4 (standard reference, not scraped)
H. Pinkham, Linear Algebra, Section 12.3 (standard reference, not scraped)

Let KK be a field, p≥1p≥1, and A∈Mp(K)A∈Mp​(K). If χA(t)=∏i<p(t−λi)χA​(t)=∏i<p​(t−λi​) in K[t]K[t], then the transfer-matrix trace series is ∑i<p(1−λix)−1∑i<p​(1−λi​x)−1