Expanded Evaluation Matrix Overview

• Expanded Evaluation Matrix is a domain-dependent framework that enhances traditional evaluations by adding explicit structural components such as latent factors, extra rows/columns, and diagnostic dimensions.
• It spans diverse applications—from language model representation and collaborative scoring to multi-criteria decision-making—demonstrating improved robustness and reduced rank reversal.
• Methodologies leveraging low-rank completion, entropy measures, and nuclear norms reveal hidden structure, offering targeted insights and actionable inference for complex systems.

An expanded evaluation matrix is a domain-dependent construct that enlarges a baseline evaluation object so that additional structure becomes explicit in the assessment itself. In the cited literature, the term denotes at least five closely related patterns: a composite representational metric built from covariance-matrix entropy and Matrix Nuclear Norm for LLMs; a partially observed model-by-item or model-by-benchmark score matrix completed under low-rank assumptions; a multi-axis diagnostic matrix that separates calibration, discrimination, business dimensions, or safety hazards; a fixed-point aggregation of trust or transmission relations across paths, blocks, or modes; and an augmented decision matrix with virtual alternatives for reducing rank reversal in multi-criteria decision-making (Vo, 2024, Fisch et al., 6 Jul 2026, Zeng et al., 22 Jun 2026, Manokhin, 5 May 2026, Benavides et al., 6 Jul 2026, Lu et al., 14 Apr 2026, Lim et al., 26 Aug 2025, Dumas et al., 2011, Guo et al., 2020, 0805.0615, Jiang et al., 10 Sep 2025).

1. Core meanings and recurring structural patterns

Across these papers, expansion is performed by adding one or more of the following: extra rows or columns, latent factors, block structure, diagnostic dimensions, field-coordinate components, or virtual alternatives. The result is not a single universal matrix type, but a recurring design principle in which a more informative or more stable evaluation object replaces a simpler baseline.

Context	Expanded object	Primary function
LLM representation evaluation	Hidden-state covariance plus entropy and Matrix Nuclear Norm	Representational diversity, redundancy, and effective dimensionality (Vo, 2024)
Collaborative or benchmark evaluation	Partially observed score matrix completed by low-rank methods	Statistical efficiency and score prediction under sparse annotation (Fisch et al., 6 Jul 2026, Zeng et al., 22 Jun 2026)
Probability and production diagnostics	Multi-axis or multi-paradigm score matrix	Failure diagnosis, model archetyping, and deployment-oriented trade-offs (Manokhin, 5 May 2026, Benavides et al., 6 Jul 2026, Lu et al., 14 Apr 2026, Lim et al., 26 Aug 2025)
Trust and transmission analysis	Matrix powers, block operators, or fixed points	Path aggregation, sequential acquisition, and high-dimensional control (Dumas et al., 2011, Guo et al., 2020)
Coding theory and MCDM	Symbol-wise field expansion or virtual alternatives	Algebraic preservation or rank-reversal mitigation (0805.0615, Jiang et al., 10 Sep 2025)

A common structural motif is that the expanded matrix is not merely a display format. It is the computational object on which aggregation, optimization, or inference is carried out. In the trust formulation, the expanded matrix is the fixed point $E$ of monoid-power iteration; in sequential transmission measurement, it is the assembled spatiotemporal operator $T$; in expanded cyclic codes, it is the $GF(q)$-level generator or parity-check matrix; in entropy-weight TODIM, it is the original decision matrix augmented by virtual alternatives; and in business or safety evaluation, it is the matrix that links metrics to dimensions, paradigms, or hazard scenarios (Dumas et al., 2011, Guo et al., 2020, 0805.0615, Jiang et al., 10 Sep 2025, Benavides et al., 6 Jul 2026, Lu et al., 14 Apr 2026, Lim et al., 26 Aug 2025).

2. Representation-space evaluation for LLMs

In "Combining Entropy and Matrix Nuclear Norm for Enhanced Evaluation of LLMs" (Vo, 2024), the expanded evaluation matrix is built from hidden-state activations. Given an input sequence $x$, hidden states are collected at a chosen layer as $H_{\text{raw}} \in \mathbb{R}^{n \times d}$, where $n$ is the number of tokens and $d$ the hidden dimension. The method then applies mean-centering,

$$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$

followed by per-token $L_2$-normalization,

$$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$

From $T$0, the empirical covariance matrix

$T$1

is computed. Its eigenvalues or singular values are normalized into a probability vector $T$2, and the matrix entropy is then

$T$3

The paper interprets $T$4 as measuring “uncertainty” or diversity in the representation space: higher $T$5 corresponds to spread-out, less redundant features, while lower $T$6 indicates compressed or redundant representations. From the same SVD, the Matrix Nuclear Norm is

$T$7

which approximates the rank of $T$8 and captures how many effective dimensions the model uses in its hidden-state space (Vo, 2024).

These two quantities are combined into the composite score

$T$9

The weighting guidance is explicit: prioritize $GF(q)$0 for representational diversity in open-ended generation, increase $GF(q)$1 for fine-grained discriminability in classification, and choose $GF(q)$2 either by a small grid search on a held-out validation set or by setting $GF(q)$3 and sweeping one parameter from $GF(q)$4 to $GF(q)$5. The computational profile is also explicit. Steps 1–3 run in $GF(q)$6, covariance formation in $GF(q)$7, and full SVD in $GF(q)$8, reducible to $GF(q)$9 with truncated SVD when $x$0. Under $x$1, the method runs effectively in $x$2, which the paper describes as practical even for $x$3 in the low-thousands (Vo, 2024).

The reported experimental findings are framed around robustness, efficiency, and ablation. On LLama-3 family models (1B→70B), GPT-4 small and medium variants, and Mistral-7B, the ranking induced by $x$4 remained consistent across different random seeds and input prompts, with $x$5 rank-swap noise. On a 3072-dimensional hidden space, truncated SVD with $x$6 produced a $x$7 speed-up over full matrix entropy and approximately $x$8 over naïve rank-based metrics. The ablation results are also central: $x$9 reduced the metric to pure entropy and lost sensitivity to representation “spread,” while $H_{\text{raw}} \in \mathbb{R}^{n \times d}$0 over-favored models with trivial redundancy; the hybrid score balanced these extremes. The paper further situates $H_{\text{raw}} \in \mathbb{R}^{n \times d}$1 alongside perplexity, downstream accuracy or F1, calibration error, and fairness or bias measures, thereby embedding the composite score in a broader multi-metric evaluation matrix (Vo, 2024).

3. Sparse score matrices, low-rank completion, and evaluation-budget reduction

A second major meaning of expanded evaluation matrix appears when evaluation is treated as matrix completion. In "CollabEval: Statistically Efficient Collaborative Model Evaluation via Matrix Completion" (Fisch et al., 6 Jul 2026), the score matrix is

$H_{\text{raw}} \in \mathbb{R}^{n \times d}$2

Only a small fraction $H_{\text{raw}} \in \mathbb{R}^{n \times d}$3 of prompts is annotated for each target model, while anchor models have dense historical scores. Under a low-rank assumption, the matrix is completed either through penalized nuclear norm minimization,

$H_{\text{raw}} \in \mathbb{R}^{n \times d}$4

or through fixed-rank factorization $H_{\text{raw}} \in \mathbb{R}^{n \times d}$5. The completed matrix then serves as a control-variate predictor. For a target model $H_{\text{raw}} \in \mathbb{R}^{n \times d}$6, the unbiased estimator is

$H_{\text{raw}} \in \mathbb{R}^{n \times d}$7

with an optional optimal control-variate coefficient

$H_{\text{raw}} \in \mathbb{R}^{n \times d}$8

The framework also derives asymptotically valid confidence intervals through a joint CLT and plug-in covariance estimation (Fisch et al., 6 Jul 2026).

The practical significance of this expanded matrix is statistical rather than descriptive. Cross-fold matrix completion provides predictions $H_{\text{raw}} \in \mathbb{R}^{n \times d}$9 for observed and unobserved entries, and those predictions are explicitly used to reduce variance while preserving unbiasedness. Empirically, across AlpacaEval 2.0, MMLU, AQA, WMT24++, and SWE-bench, the method yields a $n$0–$n$1 average reduction in $n$2 confidence-interval width compared with classical sample-mean estimators at the same annotation budget, similar reductions in MSE, effective sample fractions often $n$3–$n$4 larger than $n$5, and empirical CI coverage near the nominal $n$6 level even down to $n$7. Paired sampling over the same prompt subset further improves variance reduction in pairwise comparisons (Fisch et al., 6 Jul 2026).

A closely related construction is presented in "You Don't Need to Run Every Eval" (Zeng et al., 22 Jun 2026). There the score matrix

$n$8

contains raw scores of $n$9 models on $d$0 benchmarks. After bounded-score transformations such as logit or probit and per-column standardization, the authors report that the public $d$1 matrix, with $d$2 of cells filled, is approximately rank-2. They support this in two ways: scores hidden from the matrix are best recovered with two factors, and the first two singular values explain over $d$3 of the variation among models on shared benchmarks. BenchPress then fits a bias-decomposed ALS model in transformed space,

$d$4

with a small regularizer $d$5, and reconstructs missing entries. Held-out prediction error is reported as $d$6 raw-score points (Zeng et al., 22 Jun 2026).

BenchPress adds a confidence layer that predicts likely absolute error before observing the true score. The feature sets include ensemble spread across predictor variants and matrix-support quantities such as number of observed benchmarks for a model, number of observed models for a benchmark, and correlations with nearest peers or neighbors. This allows thresholding by estimated risk, and split-conformal scaling is used to form intervals $d$7. The same framework is used to select minimal probe subsets. A five-benchmark set $d$8 recovers the rest of a model’s public scorecard to within $d$9 points, and a cheaper five-benchmark set $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$0 attains $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$1 points (Zeng et al., 22 Jun 2026).

Taken together, these works show an expanded evaluation matrix serving as an inferential substrate: evaluation becomes a problem of reconstructing, regularizing, and uncertainty-quantifying a partially observed performance surface rather than measuring every cell directly.

4. Diagnostic and deployment-oriented evaluation matrices

In diagnostic evaluation, expansion often means decomposing a scalar notion of quality into orthogonal or semi-orthogonal axes. "The Manokhin Probability Matrix: A Diagnostic Framework for Classifier Probability Quality" (Manokhin, 5 May 2026) begins from the claim that the Brier score conflates reliability and resolution. The framework separates these with a $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$2 matrix whose horizontal axis is calibration, measured by the absolute Spiegelhalter $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$3-statistic,

$$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$4

and whose vertical axis is discrimination, measured by AUC–ROC and expected rank across datasets or folds. The four quadrants define the archetypes Eagle, Bull, Sloth, and Mole. A central theoretical statement is Proposition 1: for nondecreasing $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$5, $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$6, so monotone post-hoc calibrators cannot improve AUC. The expanded blueprint then adds class-wise calibration, sharpness, cost-sensitivity, resolution, multiclass extensions, and visualization strategies such as $$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$7 grids, radar charts, parallel coordinates, and low-dimensional embeddings (Manokhin, 5 May 2026).

"EvalLoop: A Methodology for Evaluation-Driven Iterative Improvement of Business AI Systems" (Benavides et al., 6 Jul 2026) uses expansion in a more operational sense. In the sales-briefing case study, 18 metrics are grouped into five dimensions: Structural Compliance, Content Accuracy, Hallucination Free Rate, Business Logic, and Synthesis Power. The per-dimension score is

$$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$8

with uniform weights in the case study, and the overall score is

$$\bar h = \frac{1}{n}\sum_{i=1}^n H_{\text{raw}}[i,:], \qquad H_{\text{centered}}[i,:] = H_{\text{raw}}[i,:] - \bar h,$$9

The methodology then adds failure-mode classification within weak dimensions and a structured iteration workflow in which exactly one variable is changed per run. In the reported study, dimensional diagnosis showed that $L_2$0 of hallucination failures were prompt-induced interpretation errors. A targeted prompt fix improved the best model from $L_2$1 to $L_2$2 overall, with Content Accuracy increasing by $L_2$3 percentage points and Synthesis Power by $L_2$4 percentage points. A one-time blind SME gate on 4 finalists and 16 cases reduced review burden by $L_2$5 relative to the full design (Benavides et al., 6 Jul 2026).

"AlphaEval: Evaluating Agents in Production" (Lu et al., 14 Apr 2026) formalizes a related but broader matrix. Its 94 tasks, sourced from seven companies across six O*NET domains, combine reference-driven metrics, formal logic verification, execution-based verification, rubric-based assessment, and LLM-as-a-Judge. A task-level score is

$L_2$6

the domain score is

$L_2$7

and the overall score is

$L_2$8

The matrix representation

$L_2$9

records average per-domain, per-paradigm performance. The framework couples this matrix to a requirement-to-benchmark pipeline with partner engagement, requirement elicitation, task formalization, and 3–4 iterative validation loops (Lu et al., 14 Apr 2026).

"MATRIX: Multi-Agent simulaTion fRamework for safe Interactions and conteXtual clinical conversational evaluation" (Lim et al., 26 Aug 2025) expands evaluation around safety engineering. Its safety library contains 17 Patient Input Types, 28 Expected Safe Behaviors, and 40 Hazardous Scenarios. Expected behaviors and failure modes are organized through a scenario-failure matrix

$$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$0

BehvJudge, implemented as prompt-based classification on Gemini-2.5-Pro, attains expert-level hazard detection with $$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$1 and sensitivity $$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$2 on a blinded assessment of 240 dialogues. PatBot is selected as Llama-3.3-70B at temperature $$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$3 after script-adherence, realism-ranking, and patient-preference analyses. In the 2,100-dialogue benchmark spanning 14 hazards and 10 clinical domains, Gemini-2.5-Pro achieved a mean accuracy of $$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$4, with particularly weak performance on emergency scenarios HS2 ($$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$5) and HS8 ($$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$6) (Lim et al., 26 Aug 2025).

These diagnostic matrices differ in scale and objective, but they share a common operational logic: aggregate scores are decomposed into interpretable axes or cells, and those cells are then tied to interventions, deployment trade-offs, or safety cases.

5. Trust propagation, spatiotemporal operators, and algebraic expansions

Outside model benchmarking, the term also denotes genuinely expanded algebraic operators. In "Matrix powers algorithms for trust evaluation in PKI architectures" (Dumas et al., 2011), each ordered pair $$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$7 is assigned a trust triple

$$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$8

with $$H_{\text{norm}}[i,:] = \frac{H_{\text{centered}}[i,:]}{\|H_{\text{centered}}[i,:]\|_2}.$$9 for no direct edge and $T$00 on the diagonal. Sequential aggregation and parallel aggregation are defined as two monoid laws on

$T$01

Matrix powers $T$02 then collect all paths of length $T$03, with sequential composition along edges and parallel aggregation across paths. In an acyclic graph, once $T$04 reaches the longest directed path length $T$05, the expanded evaluation matrix is the fixed point

$T$06

equivalently $T$07 under entrywise parallel aggregation. In cyclic graphs, a memory-based algorithm prevents infinite loop reuse. Complexity is $T$08 for DAGs and $T$09 in the cyclic case with edge-set memory, while long paths often contribute negligible trust and rapid convergence is reported in $T$10 or even constant steps for fixed precision (Dumas et al., 2011).

In "Sequential Transmission Matrix Evaluation Via Spatiotemporal Transmitted Modes Decomposition" (Guo et al., 2020), the expanded evaluation matrix is a high-dimensional spatiotemporal transmission operator. The continuous relation

$T$11

is discretized into a block matrix $T$12 of size $T$13, with coefficients $T$14. Spatial modes are partitioned into blocks $T$15, temporal basis functions into windows $T$16, and $T$17 is viewed as a $T$18 block matrix with blocks $T$19. Under static-medium and linear-superposition assumptions, each block is acquired sequentially while already measured blocks are held in optimized states, thereby maximizing local SNR. The paper states that $T$20 grows monotonically as more blocks are turned on. Once $T$21 is assembled, it supports focusing and image reconstruction. Reported experiments give $T$22 for sequential TM versus $T$23 for direct TM at $T$24, $T$25 at $T$26 with 64 subblocks while direct TM is computationally infeasible, and per-block SNR increasing from approximately $T$27 to approximately $T$28 over a 4-block sequence (Guo et al., 2020).

In coding theory, "On Expanded Cyclic Codes" (0805.0615) defines the expanded generator and parity-check matrices of a cyclic code over $T$29 using a $T$30-basis $T$31 and coordinate maps $T$32. If $T$33 is the original generator matrix, the expanded generator matrix

$T$34

is obtained by replacing each row $T$35 with the $T$36 coordinate rows $T$37. An analogous construction yields $T$38. The paper states that cyclic shift structure survives symbol-wise expansion, that if the original code has minimum distance $T$39 then $T$40, that designed-distance bounds carry over, that certain BCH codes become constant-weight after expansion, and that fixed-rate binary expanded Reed–Solomon codes are asymptotically “bad” (0805.0615).

These three cases share a formal property: the expanded matrix is not an auxiliary report but the explicit operator in which path composition, block acquisition, or field-coordinate structure is made computationally accessible.

6. Rank-stability, virtual alternatives, and cross-domain methodological themes

In multi-criteria decision-making, the expanded evaluation matrix is used to reduce the probability of rank reversal. "An expanded evaluation matrix for the entropy-weight TODIM method to reduce the rank reversal probability and its application in selecting energy storage technology" (Jiang et al., 10 Sep 2025) starts from an original evaluation matrix

$T$41

for real alternatives $T$42 and criteria $T$43. For each criterion $T$44, the method introduces $T$45 equally spaced virtual alternatives between

$T$46

using

$T$47

The expanded matrix $T$48 appends these $T$49 virtual rows to the original $T$50 real alternatives. Entropy weights are then recomputed over all $T$51 alternatives, and the standard TODIM dominance and global score calculations proceed unchanged over the expanded set (Jiang et al., 10 Sep 2025).

The simulation study varies the number of real alternatives $T$52, criteria $T$53, level parameter $T$54, and number of newly added alternatives $T$55. Over all $T$56 $T$57 combinations, replacing the original matrix with the expanded matrix reduced rank-reversal probability in every case. The average probability gap is $T$58, described as a $T$59 reduction on a baseline average $T$60, with standard deviation $T$61. The anti-reversal effect declines as $T$62 increases, from average $T$63 at $T$64 to $T$65 at $T$66, and correlation tests show $T$67 is positively correlated with the number of real alternatives and especially with $T$68, and negatively correlated with $T$69. The authors also report moderate inconsistency when only virtual alternatives are added, with mean consistency probability approximately $T$70, although the average number of reversed pairs per replicate is approximately $T$71 (Jiang et al., 10 Sep 2025).

The energy-storage case study makes the stabilization role concrete. With $T$72, the expanded $T$73 matrix preserves the top-ranked lithium-iron-phosphate battery $T$74 even after two new technologies are introduced; without the expanded matrix, the top choice flips from $T$75 to $T$76. Sensitivity analysis over $T$77 shows some minor pairwise reversals, but the global top alternative remains invariant (Jiang et al., 10 Sep 2025).

This body of work suggests that “expanded evaluation matrix” is best understood as a methodological family rather than a single object. In one branch, expansion improves representational diagnosis through entropy, nuclear norm, or multi-dimensional scoring (Vo, 2024, Manokhin, 5 May 2026, Benavides et al., 6 Jul 2026). In another, it converts sparse evaluation into a matrix-completion and inference problem with explicit uncertainty quantification (Fisch et al., 6 Jul 2026, Zeng et al., 22 Jun 2026). In a third, it exposes latent structure in trust networks, scattering operators, or algebraic codes (Dumas et al., 2011, Guo et al., 2020, 0805.0615). In a fourth, it stabilizes rankings by augmenting the alternative set itself (Jiang et al., 10 Sep 2025). A plausible implication is that expansion is most valuable when the original evaluation object suppresses structure that is operationally decisive: hidden redundancy, inter-model dependence, failure modes, path composition, field coordinates, or sensitivity to newly introduced alternatives.