Matrix Cores: Theory and Applications

• Matrix cores are invariant substructures in matrix powers that capture asymptotic behavior through eigencones in both nonnegative and max algebra.
• Generalized core inverses extend conventional pseudo-inversion techniques, facilitating analysis of singular and index-deficient matrices across various applications.
• Hardware matrix cores, exemplified by GPU Tensor Cores, enable high-throughput mixed-precision computation and significant speedups in AI and scientific workloads.

Matrix cores are foundational algebraic and computational structures that appear in multiple domains of mathematics and modern high-performance computing. The term encompasses the geometric and algebraic core of a matrix in nonnegative linear/max algebra, generalized core inverses in classical linear algebra, "core" covariance matrices in statistical estimation, and the hardware "matrix core" (Tensor Core) units in specialized computational accelerators. Recent research has formalized these notions, unified their properties, and extended their relevance to both theory and large-scale practical applications.

1. Classical and Max-Algebraic Cores of Matrices

The classical core of a nonnegative matrix is defined as the intersection of the positive cones generated by its powers: $$\mathrm{core}(A) := \bigcap_{k=1}^{\infty} \mathrm{span}_+(A^k)$$ In nonnegative linear algebra, this uses nonnegative linear combinations; in max algebra, it is the intersection of max-cones formed by columns of $A^k$ (Butkovic et al., 2012).

A key structural result is that the core is finitely generated and is equal to the Minkowski sum of eigencones of all matrix powers: $$\mathrm{core}(A) = \sum_{k \geq 1,\, \rho \in \Lambda(A)} V(A^k, \rho^k)$$ where $V(A^k, \rho^k)$ is the cone of nonnegative eigenvectors with eigenvalue $\rho^k$ (Butkovic et al., 2012). This sequence of cones is ultimately periodic; the period corresponds to the least common multiple of cyclicities of spectral classes or critical components. In both settings, the core acts as an invariant subcone revealed by long-term matrix action.

In max algebra specifically, the core structure is richer due to the combinatorics of critical graphs and the behavior of cyclic classes: $$\operatorname{core}(A) := \bigcap_{t \geq 1} \operatorname{span}_{\text{max}}(A^t) = \bigoplus_{t \geq 1,\, p \in \Lambda(A)} V(A^t, p^t)$$ Its extremals correspond to periodic orbits of action on eigenvectors and encode the long-term growth directions for dynamic systems (Butkovic et al., 2013). The map $x \mapsto Ax$ is surjective on the core but may fail to be bijective, which has ramifications for characterizing robust, orbit periodic, and weakly stable matrices.

2. Generalized Core Inverses

Generalized inverses including the core inverse, core-EP inverse, and DMP-inverse index the minimum requirements for matrix invertibility, range, and orthogonality (Xu et al., 2017). Recent extensions—the $(i,m)$-core inverse and $(j,m)$-core inverse—strictly generalize these concepts to arbitrary indices and matrix powers:

• $(i,m)$-core inverse:

$$X = A^D A^i (A^i)^\dagger, \quad A^m X = A^i (A^i)^\dagger$$

• $(j,m)$-core inverse:

$$X = A^D A^m (A^j)^\dagger, \quad A^m X = A^m (A^j)^\dagger$$

where $A^D$ is the Drazin inverse and $(A^i)^\dagger$ the Moore–Penrose inverse (Xu et al., 2017). These inverses yield powerful formulas based on canonical/Jordan decompositions and are well-suited for singular or index-deficient matrices.

They unify prior notions: for appropriate parameter choices, the new core inverses coincide with ordinary core, core-EP, and DMP-inverses. This generalization is central in advanced matrix analysis, index theory, and applications requiring pseudo-inversion under algebraic constraints.

3. Core Covariance Matrices and the Kronecker-Core Decomposition

In statistical estimation, the core covariance matrix arises from the unique decomposition of a general covariance structure into separable (Kronecker-structured) and residual (core) components (Hoff et al., 2022). For a covariance matrix $\Sigma$ (of vectorized matrix-variate data), the Kronecker-core decomposition is: $$\Sigma = H C H^\top, \quad H H^\top = k(\Sigma), \quad c(\Sigma) = H^{-1} \Sigma H^{-\top}$$ where $k(\Sigma)$ is the nearest separable matrix (in Stein loss), $C$ is the core residue, and $H$ is a separable square root (Hoff et al., 2022). For truly separable covariance, the core is the identity; otherwise, it encodes the non-separable structure.

This framework supports core shrinkage covariance estimation, adaptively regularizing only the non-separable component via empirical Bayes with consistent estimators, facilitating robust inference in high-dimensional or low-sample regimes. The method theoretically subsumes both full, unstructured and strictly separable estimators.

4. Matrix Cores in Stochastic Analysis and Dirichlet Forms

In infinite-dimensional probability, the core of a Dirichlet form is a dense set of functions that generates the closure of the form. For interacting Brownian motions from random matrix theory, polynomial functions on configuration spaces are shown to be a core for the associated Dirichlet forms describing Dyson and Airy point processes. This result ensures that both combinatorial (correlation function-based) and analytic (SDE-based) constructions of infinite-dimensional Markov processes yield the same stochastic dynamics (Osada et al., 2014).

5. Hardware Matrix Cores: GPU Tensor Core Formalization

Matrix cores in hardware refer to specialized computational units—Tensor Cores—in Nvidia Volta, Turing, and Ampere GPUs, developed for high-throughput, mixed-precision matrix multiplication (Valpey et al., 21 Feb 2025). The formal SMT-based model presented in (Valpey et al., 21 Feb 2025) precisely captures:

• Precision Model: Inputs in FP16, products computed in FP32 (no rounding), accumulation always FP32.
• Rounding Mode: Multiplier stage exact; output stage round-to-nearest (IEEE 754); accumulator uses truncation on significand alignment—not round-to-zero as previously thought.
• Accumulation Behavior: Non-associative addition, but accumulation order has no effect due to alignment to max exponent; importantly, intermediate sums are not normalized.

This formalization enables accurate simulation and design of algorithms for matrix cores, automatically generating inputs to distinguish hardware revision differences and assessing error propagation in mixed-precision settings.

6. Practical Applications and Matrix Core-Accelerated Computation

Matrix core hardware units (Tensor Cores) are now integral to sparse and dense matrix-matrix multiplication in scientific and AI workloads. Recent work focuses on reconciling the block-dense constraints of Tensor Cores with the irregularity of unstructured sparse matrices:

• Sparse-dense MM kernels for general matrices: cuTeSpMM (Xiang et al., 8 Apr 2025) and Libra (Shi et al., 28 Jun 2025) employ synergistic task mapping between Tensor and CUDA cores, 2D-aware tiling, operational intensity modeling, and hybrid kernel execution, delivering substantial performance gains for SpMM and SDDMM.
• Block-sparse libraries (SMaT): Achieve hardware-aligned block formats (BCSR), matrix permutation to minimize zero blocks, low-level CUDA MMA API utilization, and up to $125\times$ speedups over legacy libraries (Okanovic et al., 21 Aug 2024).
• Exact accuracy recovery: Theoretical and empirical error correction schemes, coupled with external accumulation in FP32 'round-to-nearest' arithmetic, allow Tensor Cores to deliver FP32-level matrix multiplication accuracy with high throughput (Ootomo et al., 2022).

7. Core Sets in Matrix Rings over Finite Fields

In the context of algebraic combinatorics, a subset $S \subseteq M_n(R)$ is core if its null ideal $N(S)$ forms a two-sided ideal in $M_n(R)[x]$ (Rissner et al., 7 May 2024). Asymptotic enumeration for $M_2(\mathbb{F}_q)$ reveals that almost every subset is core as $q \to \infty$, with explicit formulae quantifying noncore subset rarity. The characterization depends on invertibility and module decompositions within similarity classes.

Table: Matrix Core Concepts Across Domains

Theory/Context	Definition/Key Object	Main Property
Nonnegative/max algebra	$\bigcap_{k} \mathrm{span}_+(A^k)$	Finitely generated; sum of eigencones
Generalized inverse theory	$(i,m)$-core inverse, $(j,m)$-core inverse	Parametric, Drazin/MP-based unification
Covariance estimation	Kronecker-core decomposition, Core matrix $C$	Residual structure; adaptive shrinkage
Dirichlet forms (stochastic)	Core set of functions for form closure	Polynomial functions as core
Hardware (Tensor Core)	Matrix Core accelerator units	FP16/FP32, truncation rounding, SMT
Matrix rings over fields	Core subsets w.r.t. null ideal structure	Asymptotic ubiquity of core sets

References and Notable Papers

• Periodicity and core structure: (Butkovic et al., 2012, Butkovic et al., 2013)
• Generalized core inverses: (Xu et al., 2017)
• Kronecker-core covariance estimation: (Hoff et al., 2022)
• Dirichlet form cores in random matrices: (Osada et al., 2014)
• Formal model for hardware matrix cores: (Valpey et al., 21 Feb 2025)
• Sparse Tensor Core SpMM (cuTeSpMM, Libra, SMaT): (Xiang et al., 8 Apr 2025, Shi et al., 28 Jun 2025, Okanovic et al., 21 Aug 2024)
• FP32 accuracy recovery from Tensor Cores: (Ootomo et al., 2022)
• Counting core sets over fields: (Rissner et al., 7 May 2024)

8. Conclusion

Matrix cores unify geometric, algebraic, statistical, combinatorial, and computational hardware perspectives. Whether as intersection cones governing asymptotic dynamics, generalizations of the Moore–Penrose inverse, components in covariance decomposition, hardware-accelerated operators, or large-scale set-theoretic objects, matrix cores and their rigorous formalization are central both for deep theoretical advances and real-world computational reliability. Recent research clarifies previously misunderstood properties (such as hardware rounding modes), sharpens generalization frameworks (core inverses, core shrinkage estimators), and streamlines high-performance implementations for diverse matrix workloads.