Gradient of Characteristic Poly Coefficients

• Gradient of characteristic polynomial coefficients defines the matrix of partial derivatives, generalizing the classical adjugate via principal minors.
• It unifies fundamental identities including the Cayley–Hamilton theorem and Newton identities into a comprehensive framework for matrix invariants.
• The derived algebraic branching programs (ABPs) offer efficient computational tools, halving width and reducing size compared to previous methods.

The gradient of the coefficient of the characteristic polynomial, as developed in "On the gradient of the coefficient of the characteristic polynomial" (Ikenmeyer, 7 Nov 2025), generalizes the adjugate matrix and reveals deep structural identities relating to polynomial invariants of a matrix. The work introduces a unified framework based on the bivariate Cayley–Hamilton theorem, connecting principal minors, Newton identities, and novel algebraic branching program (ABP) constructions for the efficient computation of all coefficients of the characteristic polynomial over arbitrary commutative rings. The systematic paper of the matrix gradient and its consequences provides powerful combinatorial and algorithmic tools for linear algebra and computational complexity.

1. Definitions and Basic Properties

Let $X=(X_{ij})_{1\leq i,j\leq n}$ be an $n\times n$ matrix over a commutative ring, and define the characteristic polynomial as

$$p_X(\lambda) = \det(\lambda I - X) = \lambda^n - c_1(X)\lambda^{n-1} + c_2(X)\lambda^{n-2} - \cdots + (-1)^n c_n(X),$$

where $c_k(X)$ denotes the sum of all $k\times k$ principal minors of $X$:

$$c_k(X) = \sum_{S\subset\{1,\ldots,n\},\,|S|=k} \det X_{S,S}.$$

The gradient $\nabla_X c_k(X)$ is the $n\times n$ matrix of partial derivatives:

$$[\nabla_X c_k(X)]_{ij} = \frac{\partial c_k(X)}{\partial X_{ij}}.$$

For $k=n$, $c_n(X)=\det(X)$, and $\nabla_X \det(X)$ is the cofactor matrix; its transpose is the classical adjugate. For $k=n-1$, $c_{n-1}(X)=\operatorname{tr}(X)$ and $\nabla_X c_{n-1}(X)=I$.

2. The Bivariate Cayley–Hamilton Theorem

The bivariate Cayley–Hamilton theorem, a direct extension of the classical form, states that for $0\leq d\leq n-1$,

$$(\nabla_X c_{d+1}(X))^T = \sum_{i=0}^d (-1)^i c_{d-i}(X) X^i.$$

For $d = n-1$, this recovers the identity

$$(\nabla_X \det(X))^T = \sum_{i=0}^{n-1} (-1)^i c_{n-1-i}(X) X^i,$$

leading to the classical Cayley–Hamilton theorem:

$\sum_{i=0}^n (-1)^i c_{n-i}(X) X^i = 0.$

The proof proceeds via combinatorial expansions, comparing terms based on cycle covers and their unions, with involutive cancellation yielding equality between both sides.

3. Combinatorial Formulas for the Gradient Entries

Two principal forms describe the entries of $\nabla_X c_k(X)$:

• Minor-Sum Form:

$$[\nabla_X c_k]_{ij} = \sum_{\substack{S\subset[n],\,|S|=k,\,i,j\in S}} (-1)^{r+s} \det X_{S\setminus\{i\},\,S\setminus\{j\}},$$

where $r$ (resp. $s$) is the position of $i$ (resp. $j$) in the ordered set $S$.

• Permutation-Sum Form:

$$c_k(X) = \sum_{I\subset[n],\,|I|=k} \sum_{\pi: I \to I} \operatorname{sgn}(\pi) \prod_{a\in I} X_{a,\pi(a)},$$

yielding

$$\frac{\partial c_k(X)}{\partial X_{ij}} = \sum_{\substack{I\subset[n],\,|I|=k,\,i,j\in I}} \sum_{\pi:I\to I,\,\pi(i)=j} \operatorname{sgn}(\pi) \prod_{a\in I\setminus\{i\}} X_{a,\pi(a)}.$$

These representations interpolate between minor-based and permutation-based expansions, offering insights into both algebraic and combinatorial structure.

4. Low-Dimensional Illustrative Cases

For explicit computation and verification:

• $n=2$:
  • $c_1 = \operatorname{tr}(X)$, $\nabla c_1 = I_2$
  • $c_2 = \det(X)$, $$\nabla c_2 = \begin{bmatrix} x_{22} & -x_{21} \ -x_{12} & x_{11} \end{bmatrix}$$
• $n=3$:

$c_1 = x_{11} + x_{22} + x_{33}$, $\nabla c_1 = I_3$ $c_2 = \sum_{i<j}(x_{ii}x_{jj} - x_{ij}x_{ji})$ For example, $$\frac{\partial c_2}{\partial x_{12}} = -x_{21} - x_{31}$$.

The cases $d=1$ and $d=2$ in the bivariate formula correspond to classical Newton identities and the trace version of Cayley–Hamilton.

5. Algebraic Branching Programs for Characteristic Coefficients

The matrix recurrence derived from the bivariate Cayley–Hamilton identity enables the efficient construction of ABPs computing $c_k(X)$:

• Applying the identity as a matrix recurrence in $d$ yields a homogeneous ABP of width $O(n^2)$ and size $O(n^2 d^2)$ simultaneously computing $c_{n,0},c_{n,1},\dots,c_{n,d}$ (Corollary 2.7).
• A refined block recurrence, applied to the last row of $(\nabla c_{n,d+1})^T$, yields an ABP of width at most $n^2/2 + O(n)$ and size approximately $n^2 d /2 - d^3/6 + \ldots$, totaling for $d=n$ around $n^3/3$ (Theorem 4.1).
• This improves substantially over the Mahajan–Vinay 1997 ABP for characteristic polynomial coefficients, which has width $n^2$ and size $\sim n^3$. The new construction halves the width and reduces the total size by roughly a factor of three.
• The ABP constructions in this framework eschew the ad hoc combinatorics of "clow sequences," instead exploiting gradient-based algebraic identities for a unified approach.

6. Unified Perspective and Algorithmic Significance

The systematic paper of the gradient matrix $\nabla_X c_k(X)$ and its transpose yields a unified recurrence encapsulating the identities underlying the Cayley–Hamilton theorem, Newton’s formulas, and principal minor relationships. The approach demonstrates that all coefficients of the characteristic polynomial, not just the determinant, admit ABPs of polynomial size over arbitrary commutative rings. This yields the currently smallest known ABPs for these invariants, built using purely algebraic (rather than combinatorial) mechanisms. The framework directly resolves the open question of Mahajan–Vinay (1999) concerning the necessity of clow sequences in these constructions, establishing their non-essentiality.

7. Connections and Extensions

The combinatorial arguments in the gradient-based approach exhibit structural similarities with earlier constructions by Straubing and Zeilberger, manifesting involutive cancellations and recursion patterns familiar from those works. The recurrence unifies previously independent results (Cayley–Hamilton theorem, Girard–Newton identities, and ABP existence for characteristic coefficients) under a single algebraic and algorithmic principle. This methodological shift offers new avenues for investigating algebraic complexity and invariant theory in matrix analysis over arbitrary commutative rings.