Elementary symmetric of a sum of matrices

> Source: original issue #6. The mathematical exposition and reported results below retain the author’s wording. References to supplied programs, files and execution records describe the original submission. Verification software and executable commands remain in that issue or its linked external repositories, where provided; no code or PDF files are included in this manuscript.

A partition formula for elementary symmetric functions of a matrix sum

**We-Solve Math-3280 / TOP-3027, OPG-23298.**

**Claim and attribution.** This is a detailed literature resolution of the complete catalogue question, with a self-contained proof, an explicit integer-partition organization, reproducible verification, and a correction to equation (11) of the specified source preprint. The central identity is classical: Amitsur (1980), independently Reutenauer–Schützenberger (1987). We make no priority claim for that identity, the partition regrouping, or its consequences. The 1987 paper states the needed formula over commutative rings and credits Amitsur in its added-in-proof note [3].

1. Exact target and conventions

For an $n\times n$ matrix $C$ over a field, the elementary symmetric functions are defined by principal minors:

$$
S_j(C)=\sum_{\substack{I\subseteq[n]\\|I|=j}}\det C[I,I],\qquad S_0(C)=1.
$$

Equivalently,

$$
\det(I+tC)=\sum_{j=0}^n S_j(C)t^j.
$$

The question requests a closed expression organized by integer partitions for $S_k(A_1+\cdots+A_N)$, with $0\leq N\leq k\leq n$, involving elementary symmetric functions of ordered words in the matrices [1,2]. The formula below answers that question for every permitted input. In fact it holds over every commutative ring $R$ with identity, with arbitrary $N\geq0$ and $k\geq0$, if $S_j(C)=0$ for $j>n$.

Throughout, $M=\sum_{i=1}^N A_i$. If $N=0$, $M=0$. A zero matrix and an empty product are distinct notions: an empty scalar product equals one. These conventions handle $k=N=0$.

2. The right indexing objects: primitive cyclic words

A nonempty word is a finite ordered string $w=i_1\cdots i_d$ with letters in $[N]$. Its length is $|w|=d$ and its matrix value is

$$
A_w=A_{i_1}A_{i_2}\cdots A_{i_d}.
$$

The word is **primitive** if it is not $v^r$ for a shorter word $v$ and integer $r\geq2$. This is a property of the string, unrelated to any matrix-theoretic use of the word primitive.

Identify words that differ by a cyclic rotation. For each $d\geq1$, let $\mathcal P_d$ contain exactly one representative of each primitive rotation class of length $d$. Taking the lexicographically least rotation makes this choice explicit. Let $\mathcal P=\bigcup_{d\geq1}\mathcal P_d$.

For two letters, denoted $A,B$, the representatives through length four are:

| Length | Primitive cyclic representatives |
| --- | --- |
| 1 | $A,B$ |
| 2 | $AB$ |
| 3 | $AAB,ABB$ |
| 4 | $AAAB,AABB,ABBB$ |

Neither $AA$ nor $ABAB$ is primitive. Retaining both $AB$ and $BA$ would count the same cyclic class twice. By contrast $ABC$ and $ACB$ are generally different cyclic classes; arbitrary rearrangement of letters is not allowed.

Why cyclic invariants are well defined even for singular factors

For arbitrary square matrices $U,V$ over a commutative ring,

$$
\det(I-zUV)=\det(I-zVU).
$$

To see this without inverses of $U$ or $V$, compute the determinant of the block matrix $\begin{pmatrix}I&zU\\ V&I\end{pmatrix}$ by eliminating either identity block. The two unit triangular eliminations give the two displayed determinants. It follows that $S_j(UV)=S_j(VU)$ for every $j$. Splitting a word into a first block and its remaining block proves that every $S_j(A_w)$ is invariant under cyclic rotation. No diagonalizability, eigenvalue selection, or invertibility assumption is needed.

3. The complete formula using elementary symmetric functions of words

**Theorem 1 (classical word formula, explicit finite form).** For every $k\geq0$,

$$
\boxed{
S_k(M)=
\sum_{\substack{0\leq j_w\leq n\ (w\in\mathcal P,\ |w|\leq k)\\
\sum_w |w|j_w=k}}
(-1)^{k+\sum_wj_w}\prod_w S_{j_w}(A_w).
}
$$

Only finitely many words occur: $|w|\leq k$, with $j_w\leq\lfloor k/|w|\rfloor$. Factors with $j_w=0$ equal one. A primitive class appears once; multiplicity $j$ of that class contributes the single invariant $S_j(A_w)$, not the power $\operatorname{tr}(A_w)^j$.

Every summand is an integer-coefficient product of elementary symmetric functions of ordered matrix words. There are no denominators. Signs remain meaningful in all characteristics, including characteristic two where $-1=1$.

The condition $\sum_w|w|j_w=k$ shows that every term has total degree $k$ in the matrix entries: $S_j(A_w)$ has degree $|w|j$. This is also a useful implementation check.

4. Organization by ordinary integer partitions

Write an integer partition of $k$ as $\lambda=(1^{m_1}2^{m_2}\cdots k^{m_k})$, so $\sum_d d m_d=k$ and $\ell(\lambda)=\sum_d m_d$. For each length $d$, define

$$
H_{d,r}=
\sum_{\substack{(j_w)_{w\in\mathcal P_d}\in\{0,\ldots,n\}^{\mathcal P_d}\\
\sum_{w\in\mathcal P_d}j_w=r}}
\prod_{w\in\mathcal P_d}S_{j_w}(A_w),\qquad H_{d,0}=1.
$$

If $\mathcal P_d$ is empty, $H_{d,r}=0$ for $r>0$. These are explicitly defined finite polynomials in invariants of the individual words. Equivalently, they can be read from

$$
\sum_{r\geq0}H_{d,r}u^r
=\prod_{w\in\mathcal P_d}\left(\sum_{j=0}^n S_j(A_w)u^j\right).
$$

**Theorem 2 (requested partition expression).**

$$
\boxed{
S_k(A_1+\cdots+A_N)
=\sum_{\lambda\vdash k}(-1)^{k+\ell(\lambda)}
\prod_{d=1}^k H_{d,m_d(\lambda)}.
}
$$

For $k=0$, the unique empty partition contributes one. For $k>0$, Theorem 2 is exactly Theorem 1 regrouped by $m_d=\sum_{|w|=d}j_w$. This grouping is exhaustive and disjoint, and the sign becomes $(-1)^{k+\sum_d m_d}$. No extra multinomial factor is present: assigning $j_w$ to each distinct word class already counts each choice once.

There is a convenient alternative notation: if $B_d$ is the block diagonal matrix whose blocks are the $A_w$ for $w\in\mathcal P_d$, then $H_{d,r}=S_r(B_d)$. This is only shorthand; the finite expansion above ensures that the answer is expressed in elementary symmetric functions of words, as requested, rather than hiding a new matrix-sum problem inside the notation.

5. A self-contained proof of the determinant product

The core formal identity is

$$
\boxed{
\det(I-zM)
=\prod_{w\in\mathcal P}\det(I-z^{|w|}A_w).
}\tag{D}
$$

The infinite product is interpreted as a formal power series. For a fixed coefficient $z^k$, factors with $|w|>k$ contribute only their constant term one, so that coefficient uses finitely many factors. This is not an assertion of analytic convergence for numerical $z$.

5.1 Establish the identity in characteristic zero

Initially take generic matrices over $\mathbb Q[\mathbf a]$, where $\mathbf a$ denotes their commuting entry variables. For a fixed matrix $C$, formal differentiation of the determinant gives

$$
\frac{d}{dz}\log\det(I-zC)
=-\operatorname{tr}\bigl(C(I-zC)^{-1}\bigr)
=-\sum_{r\geq1}\operatorname{tr}(C^r)z^{r-1}.
$$

The inverse is the formal geometric series. Both logarithms have zero constant term; integrating coefficientwise gives

$$
\log\det(I-zC)=-\sum_{r\geq1}\frac{\operatorname{tr}(C^r)}r z^r.
\tag{L}
$$

Expand $M^r$ as the sum of all ordered words of length $r$. For completeness, the cyclic rotations fixing a given string form a subgroup of $\mathbb Z/r\mathbb Z$. Its generator corresponds to a divisor $d$ of $r$; the string repeats its length-$d$ prefix, and that prefix is primitive by minimality. This gives its unique primitive cyclic root and exactly $d$ distinct rotations. For a primitive cyclic class represented by $w$ of length $d\mid r$, the corresponding length-$r$ words are thus precisely the $d$ distinct rotations of $w^{r/d}$. The count is $d$, not $r$: repeating the primitive root does not create additional distinct rotations. Their matrix traces all equal $\operatorname{tr}(A_w^{r/d})$ by cyclic invariance. Hence

$$
\operatorname{tr}(M^r)
=\sum_{d\mid r}d\sum_{w\in\mathcal P_d}
\operatorname{tr}(A_w^{r/d}).
\tag{W}
$$

Substitute (W) into (L), put $r=ds$, and rearrange coefficientwise finite sums:

$$
\begin{aligned}
\log\det(I-zM)
&=-\sum_{d\geq1}\sum_{w\in\mathcal P_d}\sum_{s\geq1}
\frac{\operatorname{tr}(A_w^s)}s z^{ds}\\
&=\sum_{w\in\mathcal P}\log\det(I-z^{|w|}A_w).
\end{aligned}
$$

Formal exponentiation proves (D) over $\mathbb Q[\mathbf a]$.

5.2 Descend to every commutative coefficient ring

For a fixed degree, each side of (D) is a polynomial with **integer** coefficients in the generic matrix entries. The injection $\mathbb Z[\mathbf a]\hookrightarrow\mathbb Q[\mathbf a]$ implies equality of those integer polynomials. Specializing the entries through any ring homomorphism $\mathbb Z[\mathbf a]\to R$ proves the identity over an arbitrary commutative ring $R$, including rings with zero divisors.

This argument does not take logarithms or divide by $r$ in a positive-characteristic target ring. Those operations occur only in the rational generic proof; the conclusion is an integral polynomial identity. That distinction is essential.

5.3 Extract the coefficient and check the sign

Expand each factor as

$$
\det(I-z^dA_w)=\sum_{j=0}^n(-1)^jS_j(A_w)z^{dj}.
$$

The coefficient of $z^k$ on the left is $(-1)^kS_k(M)$. On the right it is the sum of $(-1)^{\sum j_w}\prod S_{j_w}(A_w)$ over weighted degree $k$. Multiplying by $(-1)^k$ gives Theorem 1. Section 4 then gives Theorem 2. This completes a proof of the entire requested formula.

6. Low-degree formulas and the role of proper powers

The partitions of $1,2,3,4$ give compact formulas valid for every number of summands:

$$
\begin{aligned}
S_1(M)&=H_{1,1},\\
S_2(M)&=H_{1,2}-H_{2,1},\\
S_3(M)&=H_{1,3}-H_{1,1}H_{2,1}+H_{3,1},\\
S_4(M)&=H_{1,4}-H_{1,2}H_{2,1}+H_{2,2}
+H_{1,1}H_{3,1}-H_{4,1}.
\end{aligned}
$$

For two matrices, put $a_j=S_j(A)$, $b_j=S_j(B)$, and

$$
c=\operatorname{tr}(AB),\qquad
d=\operatorname{tr}(A^2B)+\operatorname{tr}(AB^2),
$$

$$
f=\operatorname{tr}(A^3B)+\operatorname{tr}(A^2B^2)+\operatorname{tr}(AB^3).
$$

The explicit identities are

$$
\begin{aligned}
S_2(A+B)&=a_2+b_2+a_1b_1-c,\\
S_3(A+B)&=a_3+b_3+a_2b_1+a_1b_2-(a_1+b_1)c+d,\\
S_4(A+B)&=a_4+b_4+a_3b_1+a_2b_2+a_1b_3\\
&\quad-(a_2+b_2+a_1b_1)c+S_2(AB)+(a_1+b_1)d-f.
\end{aligned}
$$

For example, $H_{2,2}=S_2(AB)$ because there is only one primitive binary cyclic class of length two. Thus $ABAB$ is represented through exterior degree two of the primitive word $AB$. Treating $ABAB$ as another primitive factor would introduce an erroneous additional term. This is why powers and multiplicities must be distinguished.

7. Correcting equation (11) of the cited preprint

In **arXiv:math/0612464v5**, dated 17 September 2009, equation (11) on printed page 4 gives an expression for $S_3(A_1+A_2+A_3)$. As displayed in that specific version, it omits

$$
\boxed{+\operatorname{tr}(A_1)\operatorname{tr}(A_2)\operatorname{tr}(A_3).}
$$

A complete disproof of that displayed equation is obtained by taking $A_i=E_{ii}\in M_3(\mathbb Z)$, where $E_{ii}$ has a one at position $(i,i)$ and zeros elsewhere. Then $A_1+A_2+A_3=I_3$, so its left side is $S_3(I_3)=1$. Each $S_2(A_i)$ and $S_3(A_i)$ is zero, and every mixed product appearing in that equation is zero. Its displayed right side is therefore zero. The omitted trace product is one.

Here is the fully corrected expression, written symmetrically. Set $a_i=\operatorname{tr}(A_i)$. Then

$$
\begin{aligned}
S_3(A_1+A_2+A_3)
&=\sum_{i=1}^3S_3(A_i)
+\sum_{i\ne j}S_2(A_i)a_j+a_1a_2a_3\\
&\quad-(a_1+a_2+a_3)\sum_{i<j}\operatorname{tr}(A_iA_j)\\
&\quad+\sum_{i<j}\left[\operatorname{tr}(A_i^2A_j)+\operatorname{tr}(A_iA_j^2)\right]\\
&\quad+\operatorname{tr}(A_1A_2A_3)+\operatorname{tr}(A_1A_3A_2).
\end{aligned}
$$

This follows immediately from the degree-three formula in Section 6: the missing term is the contribution of the partition $1+1+1$ with three distinct length-one classes. The remaining terms agree with the displayed source expression. The verification program checks the corrected equality on exact matrix triples as well as the explicit witness. This correction concerns the specified arXiv version; it does not presume an audit of every later version or the journal typesetting.

8. Why the mixed ordered words cannot be discarded

Consider $A=E_{12}$ and $B=E_{21}$ in $M_2(\mathbb Z)$. Both matrices are nilpotent, so all their positive-degree elementary symmetric functions vanish. Yet

$$
A+B=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
S_2(A+B)=-1.
$$

Replacing $B$ by zero leaves all the individual invariants $S_j(A),S_j(B)$ unchanged, but changes $S_2(A+B)$ to zero. Therefore the individual characteristic polynomials of the summands cannot determine the answer. Here $\operatorname{tr}(AB)=1$ supplies exactly the missing mixed information.

Even the order of distinct mixed letters matters. In $M_2(\mathbb Z)$, take $A=E_{11}$, $B=E_{12}$, $C=E_{21}$. Then $\operatorname{tr}(ABC)=1$ while $\operatorname{tr}(ACB)=0$. Thus $ABC$ and $ACB$ cannot be merged. Cyclic rotations are legitimate; arbitrary permutations are not.

9. Comparison with the Newton trace partition formula

If the coefficient ring is a $\mathbb Q$-algebra, define

$$
p_d=\operatorname{tr}(M^d)
=\sum_{i_1,\ldots,i_d=1}^N\operatorname{tr}(A_{i_1}\cdots A_{i_d}),
\qquad
z_\lambda=\prod_d d^{m_d(\lambda)}m_d(\lambda)!.
$$

Expanding the exponential in the formal logarithm identity also gives the familiar partition formula

$$
S_k(M)=\sum_{\lambda\vdash k}
\frac{(-1)^{k-\ell(\lambda)}}{z_\lambda}
\prod_d p_d^{m_d(\lambda)}.
$$

Indeed, $\det(I+tM)=\exp(\sum_{d\geq1}(-1)^{d-1}p_dt^d/d)$; selecting $m_d$ terms from each exponential contributes $1/(d^{m_d}m_d!)$. This proves the formula directly. It is valid over a field of characteristic greater than $k$ as well, because all denominators used at degree $k$ are then invertible.

It cannot simply be evaluated when those denominators vanish. The issue is stronger than notation: in characteristic two, $I_2$ and $0_2$ have identical traces of every positive power, all zero, but their determinants are one and zero. Hence all power traces do not determine $S_2$ in that characteristic. The word-elementary formula retains the information that trace-only expressions lose and avoids dividing in the target ring.

10. Mixed coefficients and a multivariable refinement

Introduce commuting scalar variables $x_1,\ldots,x_N$. For a word $w$, let $\nu_i(w)$ count occurrences of letter $i$ and put $x^{\nu(w)}=\prod_i x_i^{\nu_i(w)}$. Substituting $x_iA_i$ into (D) gives

$$
\det\left(I-\sum_i x_iA_i\right)
=\prod_{w\in\mathcal P}\det\left(I-x^{\nu(w)}A_w\right).
$$

This is a formal identity graded by total degree. Consequently the coefficient of $x^\alpha$ in $\det(I+\sum_i x_iA_i)$ is

$$
\sum_{\substack{(j_w)\geq0\\\sum_wj_w\nu(w)=\alpha}}
(-1)^{|\alpha|+\sum_wj_w}\prod_w S_{j_w}(A_w).
$$

For $\alpha=(1,1,1)$ the coefficient is

$$
a_1a_2a_3-a_1\operatorname{tr}(A_2A_3)
-a_2\operatorname{tr}(A_1A_3)-a_3\operatorname{tr}(A_1A_2)
+\operatorname{tr}(A_1A_2A_3)+\operatorname{tr}(A_1A_3A_2).
$$

This isolates the three-letter contribution responsible for the correction above. The refinement records exactly how often each summand is used rather than only the total degree; it is another direct specialization of the classical identity, not a separate novelty claim.

11. Finite implementation, costs, and boundary checks

To calculate through degree $K$:

1. Enumerate primitive words of lengths at most $K$ and retain one least cyclic rotation each.
2. Compute each ordered product $A_w$ and its elementary symmetric functions through $\min(n,\lfloor K/|w|\rfloor)$.
3. Either multiply the factors $\sum_j(-1)^jS_j(A_w)z^{j|w|}$ modulo $z^{K+1}$, or form the $H_{d,r}$ and sum over integer partitions.
4. Compare with principal-minor coefficients of $M$ using exact arithmetic.

The reference implementation deliberately uses transparent enumeration. It is a structural expansion, not a claim of faster characteristic-polynomial computation. The number of primitive length-$d$ classes is

$$
L_d=\frac1d\sum_{e\mid d}\mu(e)N^{d/e},
$$

where $\mu$ is the Möbius function. To derive the count, partition all $N^r$ length-$r$ strings by their primitive root class: $N^r=\sum_{d\mid r}dL_d$. Möbius inversion gives the formula. The program independently checks its necklace enumeration against these counts.

For $N\geq1$, there are exactly $N^k$ multiplicity assignments in Theorem 1 before evaluating or combining terms when $k\leq n$. One elementary derivation uses the scalar generating product $T(z)=\prod_d(1-z^d)^{-L_d}$. Its logarithm has coefficient $(\sum_{d\mid r}dL_d)/r=N^r/r$, so $T(z)=(1-Nz)^{-1}$. The coefficient counts the weighted assignments. For $N=k=0$, there is one empty assignment. This explains why a closed expression need not be a polynomial-size expansion.

The boundary cases are explicit:

- $k=0$: the all-zero multiplicity assignment gives one.
- $N=0,k>0$: there is no positive-degree assignment, giving zero.
- $N=1$: the sole primitive class is the one-letter word, so the formula reduces to $S_k(A_1)$.
- $k>n$: the formula remains valid and its terms cancel to zero; these are consistency identities, not extra coefficients of a larger characteristic polynomial.
- Singular, zero, repeated, commuting and noncommuting summands all satisfy the same formula.
- Small positive characteristic and zero divisors require no special case in the division-free formula.

12. Verification and remaining scope

The accompanying `verify3027.py` uses exact arithmetic and compares direct principal minors, the truncated determinant product, and the integer-partition regrouping. All three methods agreed in the executed run: **513 cases, with 3,519 coefficients compared per method**.

| Test family | Cases | Coefficients per method |
| --- | ---: | ---: |
| Every pair of binary $2\times2$ matrices | 256 | 1,792 |
| Seeded integer matrices, dimensions through five, up to three summands | 60 | 420 |
| Empty alphabet, one summand, zero degree and empty matrix boundaries | 24 | 96 |
| Quotient rings modulo $2,3,5,4,6,8$ | 144 | 1,008 |
| The extension field $\mathbb F_4=\mathbb F_2[x]/(x^2+x+1)$ | 27 | 189 |
| Explicit mixed-word and word-order witnesses | 2 | 14 |
| Total | 513 | 3,519 |

The binary suite alone checked 1,024 coefficients beyond the matrix dimension, where cancellation must give zero. Supplementary checks compared 120 determinants by independent Bareiss and permutation formulas, checked 24 primitive-necklace counts and 75 cyclic rotations with singular factors, evaluated 40 generated formal expansions, and verified the corrected source equation on 120 seeded triples. The source-equation witness returned displayed right side zero, actual value one, missing term one. The main seed is 3027; the formal-expansion evaluation seed is 12345. Full configuration, formal terms and exact results are in `verification_results.json`.

These finite checks supplement the proofs and detect implementation errors; they do not establish the unrestricted identities by themselves. Integer determinants are computed before reducing modulo composite moduli, and extension-field determinants use a division-free permutation expansion. Thus these tests do not assume that division is available in rings with zero divisors.

Two separate same-model research agents independently checked the primitive-word indexing, signs, cyclic invariance, generic integral descent, and source equation. A third same-model agent implemented and ran the verification. This is model-assisted review; no independent human proof verification or formal proof-assistant certification is asserted.

The complete formula request is answered. We do not claim an optimal minimal expression, an algorithmic speedup, or a new general invariant-theory result. The source correction is confined to equation (11) of arXiv version 5. The general identity was established before the source question was posted, so the appropriate contribution classification is **Literature report**, with a complete mathematical resolution and a verifiable source correction.

References and inspection scope

1. R. S. Costas-Santos (`rscosa`), **Elementary symmetric of a sum of matrices**, Open Problem Garden, posted 8 December 2008. Original problem. Exact range and partition request inspected.
2. R. S. Costas-Santos, **On the elementary symmetric functions of a sum of matrices**, arXiv:math/0612464v5, 17 September 2009. Sections 3–5 inspected, especially equation (11) on printed page 4 and the word/partition request on printed pages 7–8. The catalogue cites the journal appearance in Journal of Algebra, Number Theory: Advances and Applications 1 (2009), 99–112; the correction here is about the inspected arXiv version.
3. Christophe Reutenauer and Marcel-Paul Schützenberger, **A Formula for the Determinant of a Sum of Matrices**, Letters in Mathematical Physics 13 (1987), 299–302, DOI 10.1007/BF00401158. Author-hosted full paper. The theorem, cyclic-word definitions, determinant product proof and added-in-proof attribution inspected. The note credits S. A. Amitsur, **On the characteristic polynomial of a sum of matrices**, Linear and Multilinear Algebra 8 (1980), 177–182; that original 1980 full text was not inspected and is credited through the inspected 1987 paper.
4. **We-Solve / UnsolvedMath TOP-3027**, also displayed as Math-3280, OPG-23298. Statement, research prompt. Exact statement and prompt inspected; prompt listed no previous contributions. Credit: Ulam AI, the UnsolvedMath contributors and We-Solve; catalogue material under CC BY 4.0.

Provenance and licences

Requested and submitted for Gaven (`@gavenren`), whose role was prompting and authorizing submission. Research, exposition, mathematical review, source inspection and verification software were produced by OpenAI Codex (GPT-6; exact serving revision and reasoning setting unknown), including separate same-model agents. No human mathematical authorship or independent human checking is asserted.

Original exposition and results files: CC BY 4.0. Original software: Apache-2.0. Classical results and source material retain the attribution above. Referenced papers are linked, not redistributed.

Reproducible reference software

Save as `verify3027.py` and run `python3 verify3027.py` with Python 3.10 or later, assertions enabled. No third-party dependencies. The program regenerates the full JSON results and formal expansions. Source: Apache-2.0.

<details>
<summary>verify3027.py — complete exact implementation and verification</summary>

> Verification software and executable commands are external to this manuscript. See the original submission’s reproducibility section for this material.

</details>

Attribution and references from the original submission

Who did the work

LLM only (people only prompted or submitted)

LLMs used

Other (named below)

Other LLMs

OpenAI, Codex (GPT-6; exact serving model revision unknown)

LLM versions, settings and roles

OpenAI Codex (GPT-6) | version: unknown | reasoning: unknown | role: primary research, source checks, mathematical exposition and submission; two separate same-model agents independently investigated and audited the formula and source correction; a third same-model agent implemented and executed exact verification. All agents used the inherited model; no other model was knowingly used.

People and roles

Gaven (@gavenren): requested another deeply investigated problem and authorized submission. No human mathematical authorship, mathematical edits or independent human proof checking is asserted.

References

1. R. S. Costas-Santos, Elementary symmetric of a sum of matrices, Open Problem Garden (8 December 2008): https://www.openproblemgarden.org/op/elementary_symmetric_of_a_sum_of_matrices . Exact question and range inspected.
2. R. S. Costas-Santos, On the elementary symmetric functions of a sum of matrices, arXiv:math/0612464v5 (17 September 2009): https://arxiv.org/pdf/math/0612464v5 . Sections 3–5 inspected, especially Eq. (11), printed p.4, and the general request, pp.7–8. Correction is specifically to this version.
3. C. Reutenauer and M.-P. Schützenberger, A Formula for the Determinant of a Sum of Matrices, Letters in Mathematical Physics 13 (1987), 299–302, DOI 10.1007/BF00401158: https://reutenauer.math.uqam.ca/wp-content/uploads/2024/04/Determinant-of-a-sum.pdf . Full theorem, proof and added-in-proof note inspected. Its note credits S.A. Amitsur, Linear and Multilinear Algebra 8 (1980), 177–182; original 1980 full text not inspected and credited through this checked source.
4. We-Solve / UnsolvedMath TOP-3027, Math-3280, OPG-23298: https://github.com/We-Solve/research/blob/main/problems/b8/prb_b8fd80d827ea4ad186a62c524b01a848/statement.tex and https://github.com/We-Solve/we-solve.github.io/blob/main/prompts/TOP-3027.txt . Exact statement and prompt inspected; prompt listed no earlier contributions. Catalogue credit: Ulam AI, UnsolvedMath contributors and We-Solve, CC BY 4.0.

Original contributor disclosures

Source: https://github.com/We-Solve/research/issues/6

Who did the work

LLM only (people only prompted or submitted)

LLMs used

Other (named below)

Other LLMs

OpenAI, Codex (GPT-6; exact serving model revision unknown)

LLM versions, settings and roles

OpenAI Codex (GPT-6) | version: unknown | reasoning: unknown | role: primary research, source checks, mathematical exposition and submission; two separate same-model agents independently investigated and audited the formula and source correction; a third same-model agent implemented and executed exact verification. All agents used the inherited model; no other model was knowingly used.

People and roles

Gaven (@gavenren): requested another deeply investigated problem and authorized submission. No human mathematical authorship, mathematical edits or independent human proof checking is asserted.

A division-free partition formula for elementary symmetric functions of a sum of matrices

OpenAI GPT 6.1 Sol (Max reasoning)

This research was conducted under the direction and oversight of Wentao Li.

12 October 2026

Abstract

We give a finite sum indexed by integer partitions of $k$ for $S_k(A_1+\cdots+A_N)$, using only elementary symmetric functions of ordered matrix words. It applies over every field, including positive characteristic, with no assumption that the matrices commute. The answer organizes the established Amitsur cyclic-word formula by the total degrees of its selected invariants. A self-contained proof and exact checks are supplied.

Original target and established context

For $A\in M_n(F)$, where $F$ is any field, set \[ S_j(A)=\sum_{\substack{I\subseteq\{1,\ldots,n\}\\|I|=j}} \det A[I,I],\qquad S_0(A)=1. \] Thus $S_n(A)=\det A$, $S_1(A)=\operatorname{tr} A$, and \begin{equation}\label{wsref-0-64-65-74-2d-67-65-6e-65-72-61-74-69-6e-67} \det(I_n-tA)=\sum_{j=0}^n(-1)^jS_j(A)t^j. \end{equation} For convenience set $S_j(A)=0$ for $j>n$. Determinants of zero-sized matrices and empty products are $1$.

The recorded target [catalogue,costas] is to find a closed expression organized by integer partitions for \[ S_k(A_1+\cdots+A_N),\qquad 0\leq N\leq k\leq n, \] in matrix quantities built from the $A_i$, permitting elementary symmetric functions of their ordered words. Every matrix tuple over every field is included; the empty sum is the zero matrix, and the empty partition is allowed when $k=0$. No commutativity hypothesis is present.

Costas-Santos [costas, §§1,5] gives low-degree formulas and asks for a general closed expression involving partitions and elementary symmetric functions of words. The arXiv record dates the first submission to 16 December 2006 and version 5 to 17 September 2009; the later date displayed by the HTML conversion is not a new mathematical publication.

Amitsur's 1980 paper [amitsur] addresses characteristic coefficients of sums in terms of coefficients of words. We consulted its publisher abstract and bibliographic record, but could not access its full text. The full primary paper of Reutenauer and Schützenberger [rs] was visually inspected: its theorem on p. 300 and equations $6$--$7$ on p. 301 give the cyclic-word formula over a commutative ring; its p. 302 addendum credits Amitsur. The reference request [mo] was a bibliographic lead. The present answer applies this established mathematics to the exact recorded target and gives its partition organization; no novelty of the underlying identities is claimed. This does not revise the catalogue's original open source label.

Primitive words and the integral product identity

A word $w=(i_1,\ldots,i_d)$ in the alphabet $\{1,\ldots,N\}$ has length $|w|=d$ and matrix value $A_w=A_{i_1}\cdots A_{i_d}$, in precisely that order. A nonempty word is primitive if it is not a power $v^r$ with $r>1$. Identify words only under cyclic rotation, and choose the lexicographically least representative of each primitive cyclic class. Let $\mathcal C_N$ be these representatives and $\mathcal C_{N,k}=\{w\in\mathcal C_N:|w|\leq k\}$. These sets depend only on the alphabet, not on the matrix entries. When $N=0$ there are no nonempty words.

The finite partition expression

Write $\lambda\vdash k$ for an integer partition $\lambda=(\lambda_1\geq\cdots\geq\lambda_\ell>0)$ of $k$. For each such partition let $D_\lambda(N,n)$ be the following finite, explicitly enumerable set of tuples: \begin{equation}\label{wsref-0-64-65-63-6f-72-61-74-69-6f-6e-73} \begin{split} D_\lambda(N,n)=\bigl\{((w_1,r_1),\ldots,(w_\ell,r_\ell)): & w_h\in\mathcal C_{N,k},\quad 1\leq r_h\leq n,\quad r_h|w_h|=\lambda_h;\\ &w_h\ne w_j\ (h\ne j);\\ &w_h<w_{h+1}\text{ whenever }\lambda_h=\lambda_{h+1}\bigr\}. \end{split} \end{equation} The last comparison uses lexicographic order and fixes the order inside each block of equal parts. For the empty partition, $D_\varnothing(N,n)$ contains the empty tuple. An empty inner sum is $0$.

This is a closed finite expression rather than a recurrence: for each part $s$, enumerate primitive representatives of lengths $d\mid s$ with $d\leq k$, then take $r=s/d\leq n$ and enforce the displayed distinctness and ordering rules. Every scalar invariant in the answer is an $S_r$ of an ordered word in the input matrices. Each part records its total degree $r|w|$ in those matrices. Cyclic rotation is allowed; reversal and other permutations of letters are not identified.

Boundary cases.

For $k=0$ the only partition and decoration are empty, so the answer is $1$; within the recorded range this forces $N=0$. For $N=0$ and $k>0$ every decoration set is empty, giving $0$. For $N=1$ the only primitive representative is the single letter $1$: only $\lambda=(k)$ contributes when $k>0$, giving $S_k(A_1)$ with sign $(-1)^{2k}=1$. The case $k=n$ is the required determinant formula. If $n=0$ is permitted, the sole recorded case is again $N=k=0$. Singular matrices and zero input matrices need no separate hypothesis.

The characteristic-zero trace partition formula

For comparison, when $F$ has characteristic zero define \[ p_j=\sum_{(i_1,\ldots,i_j)\in\{1,\ldots,N\}^j} S_1(A_{i_1}\cdots A_{i_j}) =\operatorname{tr}\left(\left(\sum_i A_i\right)^j\right), \qquad z_\lambda=\prod_{j\geq1}j^{m_j(\lambda)}m_j(\lambda)!. \] Here $m_j(\lambda)$ is the multiplicity of the part $j$ and $\ell(\lambda)=\sum_jm_j(\lambda)$. The familiar Newton--Girard partition expression is \begin{equation}\label{wsref-0-74-72-61-63-65-2d-70-61-72-74-69-74-69-6f-6e} S_k\left(\sum_i A_i\right)= \sum_{\lambda\vdash k} \frac{(-1)^{k-\ell(\lambda)}}{z_\lambda} \prod_{j\geq1}p_j^{m_j(\lambda)}. \end{equation} Indeed, substitute $-t$ in (log-determinant) and expand \[ \det(I+tB)=\prod_{j\geq1} \exp\left(\frac{(-1)^{j-1}p_j}{j}t^j\right). \] Choosing multiplicity $m_j$ supplies the denominator $j^{m_j}m_j!$ and sign $(-1)^{\sum_j(j-1)m_j}=(-1)^{k-\ell(\lambda)}$. This proves (trace-partition) without commuting any matrix letters. Equation (partition-formula) supplies the arbitrary-characteristic answer. For example, over $\mathbb F_2$ the matrices $I_2$ and $0_2$ have identical traces of every positive power, but their $S_2$ values are $1$ and $0$. Ordinary traces of words alone cannot distinguish this one-letter example; higher $S_r$ are essential.

Examples and checks of order, signs, and repetition

For $n\geq2$ the degree-two specialization is \begin{equation}\label{wsref-0-64-65-67-72-65-65-2d-74-77-6f} S_2\left(\sum_iA_i\right) =\sum_i S_2(A_i)+\sum_{i<j} \left(S_1(A_i)S_1(A_j)-S_1(A_iA_j)\right). \end{equation} For $A=E_{12}$ and $B=E_{21}$ in dimension $2$, $AB=E_{11}$ and $BA=E_{22}$, so the pair does not commute. Equation (degree-two) gives $S_2(A+B)=-1$, agreeing with the determinant of $\left(\begin{smallmatrix}0&1\\1&0\end{smallmatrix}\right)$; over $\mathbb F_2$ this value is $1$.

For two letters the primitive representatives through length $4$ are \[ A,\ B;\quad AB;\quad AAB,\ ABB;\quad AAAB,\ AABB,\ ABBB. \] In particular $ABAB$ is periodic, and is accounted for by $S_2(AB)$ in the factor belonging to $AB$. For $n\geq4$ the formula gives \begin{equation}\label{wsref-0-64-65-67-72-65-65-2d-66-6f-75-72} \begin{aligned} S_4(A+B)={}&\sum_{\substack{a,b\geq0\\a+b=4}}S_a(A)S_b(B) -S_1(AB)\sum_{\substack{a,b\geq0\\a+b=2}}S_a(A)S_b(B)\\ &+\left(S_1(A^2B)+S_1(AB^2)\right)\left(S_1(A)+S_1(B)\right)\\ &-S_1(A^3B)-S_1(A^2B^2)-S_1(AB^3)+S_2(AB). \end{aligned} \end{equation} This recovers the source's fourth-degree invariant language in every characteristic.

Order beyond cyclic rotation matters already for $N=k=n=3$. Take $A_1=E_{12}$, $A_2=E_{23}$, $A_3=E_{31}$. Then $S_1(A_1A_2A_3)=1$ and $S_1(A_1A_3A_2)=0$. These are distinct primitive cyclic classes. In degree $3$ they are the only possibly nonzero terms of (partition-formula); their sum is $1$, which equals $S_3(A_1+A_2+A_3)$ over every field.

Verification, scope, and contributions

Exact computation.

Run python3 scripts/check_matrix_sum_3027.py from the repository root. The checker uses Python 3.13.2, its standard library, and seed $3027$. Its oracle sums principal minors using the Leibniz determinant, separately from both the decorated-partition sum and the primitive-word product. All $2293$ coefficient cases and $469$ auxiliary checks passed. The comparisons cover integer examples through dimension $6$ and every $0\leq N\leq n$; rational examples; finite fields $\mathbb F_2,\mathbb F_3,\mathbb F_5,\mathbb F_7$ through dimension $5$; every one of the $256$ ordered pairs of $2\times2$ matrices over $\mathbb F_2$; and small generic polynomial pairs in dimensions $2$ and $3$ with respectively $8$ and $18$ independent variables. Additional examples over $\mathbb Z/4,\mathbb Z/6,\mathbb Z/8$ test ring specialization. Some checks extend past degree $n$, where coefficients must vanish. The auxiliary checks cover primitive-word counts, periodic trace multiplicities, cyclic invariance, and the trace obstruction. The program also checks (trace-partition) over the rationals.

Full-solution scope declaration.

Theorem Theorem (Closed partition formula) answers the entire recorded target: an explicit finite integer-partition expression in the permitted invariants, for every matrix tuple, every field, and every $0\leq N\leq k\leq n$, including all boundary cases. Lemmas Lemma (Cyclic invariance) and Lemma (Amitsur product, with integral specialization) prove the needed identities; the proof uses only elementary determinants, formal series, finite cyclic orbit counting, and specialization of integer polynomials. The established Amitsur and Newton--Girard machinery is attributed and re-proved here. There is no asserted remaining mathematical gap for this target. The separate positivity conjecture in the source is not part of catalogue 3027's recorded question and is not addressed by this answer.

Process and attribution.

OpenAI GPT 6.1 Sol (Max reasoning) generated the proof, manuscript, and verification code. Research directed and overseen by Wentao Li. No other model was used. Filled mandatory Prompts A and B, draft and audit snapshots, source-access notes, and the exact output report are retained in the local research directory research/matrix-sum-3027-2026-10-12/, outside this repository. The checker is included with this contribution; those research notes are not asserted to have been publicly posted. The audit is a same-model review. No independent human proof verification or theorem-prover formalization has been reported; the proofs remain unverified in those senses. Computation does not replace the general proof.

Catalogue provenance.

The target and identity metadata derive from ULAM.AI and the UnsolvedMath Contributors [catalogue], under CC BY 4.0. This separately attributed attempt adds the formula, proof, and checks; it preserves the imported target and source status. Source/literature and repository collision checks were performed on 12 October 2026. No earlier 3027 attempt or matching pull request was found in the checked snapshot and searches; this is not a claim of an exhaustive historical literature search or new mathematical priority.

Completion Estimate

COMPLETION: 100% Theorem Theorem (Closed partition formula) supplies the requested closed partition expression for all recorded fields, matrix tuples, parameter values, and boundary cases. This is a full-solution claim for the recorded target using established mathematics, subject to the verification limits stated above.

References

• [catalogue] ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3027 (OPG-23298). https://huggingface.co/datasets/ulamai/UnsolvedMath. CC BY 4.0: https://creativecommons.org/licenses/by/4.0/. Recorded statement: attacks/open_problems/top_problems/definitions/3027.tex, repository snapshot 1397a8c7f75e48a6343b0afe338390b365d344df.
• [costas] R. S. Costas-Santos, On the elementary symmetric functions of a sum of matrices, arXiv:math/0612464v5 (17 September 2009), especially §§1,5. https://arxiv.org/abs/math/0612464; https://arxiv.org/html/math/0612464v5.
• [amitsur] S. A. Amitsur, On the characteristic polynomial of a sum of matrices, Linear and Multilinear Algebra 8 $1980$, 177--182. https://doi.org/10.1080/03081088008817315. Publisher abstract consulted; full text not accessed.
• [rs] C. Reutenauer and M.-P. Schützenberger, A formula for the determinant of a sum of matrices, Letters in Mathematical Physics 13 $1987$, 299--302. https://doi.org/10.1007/BF00401158. Primary scanned copy, all four pages inspected: https://igm.univ-mlv.fr/~berstel/Schutzenberger/Travaux/A/1987-2FormulaDeterminantMathPhys.pdf.
• [mo] P. Hartwig, Reference for a formula expressing the characteristic polynomial of a sum of endomorphisms, MathOverflow question 70231 (13 July 2011), with E. Formanek's answer (15 July 2011). https://mathoverflow.net/questions/70231. Consulted as a reference lead, not used as proof of the identity.