{"id":"d0431ca3-b200-4d47-8d67-2a130426994f","arxiv_id":"1908.03795","paper_version":4,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The eigenvector-eigenvalue identity, which gives squared eigenvector components from matrix and minor eigenvalues, is a classical result with a long history of rediscovery, now compiled with proofs.","lead":"This survey documents a linear algebra identity that expresses the squared entries of an eigenvector through eigenvalues of the original Hermitian matrix and of its principal minors. It traces the identity back to 1834, collects dozens of rediscoveries, and presents several proofs, making it a useful reference for mathematicians and physicists.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 2.3's coordinate-free proof is invalid: identity (22) is false; the eigenvector-eigenvalue identity itself remains true via the other proofs.","rationale":"The reader's weakest assumption concerned the historical completeness of the crowdsourced bibliography. That is a reasonable but non-load-bearing concern because the paper explicitly hedges: 'we are not able to guarantee that there is not an even earlier place in the literature' (Section 5). My review instead found a concrete mathematical error in one of the survey's proofs. The central identity is true and is established by at least two independent correct arguments (the adjugate proof in §2.1 and the Cramer-rule proof in §2.2), so the main theorem is not threatened. However, the paper claims 'a number of proofs' and the coordinate-free proof in §2.3 is invalid because it relies on the false identity (22). Since this is a survey whose value includes presenting correct proofs, a conditional accept is appropriate: the flaw is isolated and easily fixable (either correct the argument or replace Lemma 11 with a direct coordinate proof). This is not a rejection because the theorem and the historical contribution stand. The concrete counterexample is simple enough to settle the issue immediately, so the condition is minor.","tokens_in":24494,"tokens_out":14023,"duration_ms":143014,"concrete_test":"Verify (22) numerically: let n=3, T=diag(0,2,3), v=e1, f=g=e2, and compute both sides with any consistent Hodge-star convention. The left side is 0 because (f,v)=0. The right side is (T(∗f),∗f), which equals ±3 (or at least nonzero). If this calculation reproduces a nonzero RHS, (22) is false and the proof of Lemma 11 must be repaired or removed. (The main theorem can still be checked independently via the adjugate proof, e.g., by directly verifying (4) for the same T and j=1.)","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The coordinate-free proof (Section 2.3, Lemma 11) attempts to establish (20) via the stronger identity (22): (f,v)Δ_T(v)(v,g) = (T(∗f),∗g) for all f,g. The proof asserts that if f⊥v then ∗f is a wedge product of v with an (n−2)-vector, 'and hence T(∗f) vanishes.' That implication is false: for T extended as a derivation, T(v∧η)=v∧Tη, which generally is nonzero. Concretely, take n=3, T=diag(0,2,3), v=e1, f=g=e2. Then (f,v)=0 so the left side of (22) is 0. With the standard Hodge star (orientation e1,e2,e3), ∗f = ±e1∧e3, so T(∗f)=±3 e1∧e3. The inner product (T(∗f),∗g) equals ±3, not 0. Thus (22) is false. The lemma's conclusion (20) is nevertheless correct (it is just the identity (2) after normalization), and the paper's adjugate and Cramer-rule proofs are valid. The flaw is therefore not fatal to the main theorem, but the survey's statement that it provides a valid coordinate-free proof, and the reader's assessment that all proofs are correct, are inaccurate.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper surveys the identity |v_{i,j}|^2 ∏_{k≠i}(λ_i(A)-λ_k(A)) = ∏_{k=1}^{n-1}(λ_i(A)-λ_k(M_j)) for Hermitian (and more generally normal or diagonalizable) matrices, where M_j is the minor obtained by deleting row and column j. It presents several proofs (adjugate, Cramer's rule, coordinate-free exterior algebra, perturbation theory, Cauchy-Binet, and Schur complement), a collection of consistency checks, a generalization to principal minors, and an extensive historical account tracing occurrences of the identity from Jacobi (1834) through many independent rediscoveries. The paper also discusses the sociology of the identity's weak diffusion and proposes the name \"eigenvector-eigenvalue identity.\"","tokens_in":24783,"tokens_out":16046,"duration_ms":169926,"significance":"If the claims are correct, this is a valuable survey: the main identity is simple, useful, and surprisingly scattered across many communities, and the paper provides several rigorous proofs, a careful historical reconstruction, and a reasoned discussion of why such an elementary identity remained obscure. The numerical verification code and the explicit example are useful complements. The central Theorem 1 is correct and is supported by multiple independent proofs that do not rely on each other. However, the claimed generalization in Section 2.7 is false as stated, which is a load-bearing defect in one of the paper's advertised contributions; the paper cannot be accepted without correcting or removing that generalization.","major_comments":[{"comment":"Proposition 17 is false as stated for complex normal matrices. For example, take n=2, A=diag(λ1,λ2), U=diag(i,-i), so that A=UDU* with D=A, and take I=J=K={1}. The left-hand side equals det M_{{2},{2}}(U) · det M_{{2},{2}}(U) · (λ2-λ1) = i·i·(λ2-λ1) = -(λ2-λ1), while the right-hand side equals det M_{{1},{1}}(A-λ1 I_2) = λ2-λ1. This is a contradiction. The proof is also internally inconsistent: the displayed identity (24), det M_{J,I}(U) = det M_{J^c,I^c}(U) det U, is false in this same example (the left side is -i and the right side is i), whereas the derivation that follows actually yields det M_{J,I}(U) = det U · det M_{I^c,J^c}(U*). The subsequent claim det M_{I,K}(U*) = det M_{K,I}(U) ignores complex conjugation. The authors should correct the statement (for instance by inserting the appropriate phase factors or by restricting to real orthogonal U) and repair the proof, or remove this generalization from the survey.","section":"Section 2.7, Proposition 17 and Eq. (24)"},{"comment":"The proof of Lemma 11 is not self-contained as written because the phrase \"the operator T is extended to ⋀^k C^n in the usual fashion\" is ambiguous. If one uses the derivation convention T(x∧η)=Tx∧η+x∧Tη, then the assertion that T(*f) vanishes when f⊥v is false; for T=diag(0,2,3), v=e1, f=g=e2, one has T(*f)=±3 e1∧e3, so Eq. (22) fails. If, as is standard in linear algebra, one uses the induced exterior power map v1∧...∧vk ↦ Tv1∧...∧Tvk, then the step Tv∧Tη=0 is valid and the proof goes through. The authors should state this convention explicitly so that the proof is verifiable by the reader.","section":"Section 2.3, Lemma 11 and Eq. (22)"}],"minor_comments":[{"comment":"Please add one sentence defining the induced action of T on ⋀^k C^n; this would remove the ambiguity discussed in the major comment and prevent a natural misreading.","section":"Section 2.3, after Lemma 11"},{"comment":"The identity det M_{I,K}(U*) = det M_{K,I}(U) should be replaced by the correct relation involving complex conjugation, or the proof should be restricted to the real orthogonal case where conjugation is harmless.","section":"Section 2.7, proof of Proposition 17"},{"comment":"The sentence \"Jacobi's formula give us the need relationships between the eigenvalues...\" contains grammatical errors; it should read \"Jacobi's formula gives the needed relationships...\".","section":"Section 1, consistency check (x)"},{"comment":"The phrase \"For sake of notation let use permutation symmetry to set i=n\" should read \"For sake of notation, let us use permutation symmetry to set i=n.\"","section":"Section 2.2, paragraph after Eq. (18)"},{"comment":"The phrase \"An alternate approach way to arrive at (2) from (14)\" should be \"An alternate way to arrive at (2) from (14)\".","section":"Section 2.2, before Eq. (17)"}],"recommendation":"major_revision","confidential_remarks":"The core identity and most of the proofs are sound, and the historical survey is careful and appropriately hedged. The main obstacle is Section 2.7: Proposition 17 is false as stated for complex normal matrices, and the proof contains an incorrect display and an incorrect identification. This is fixable by correcting the statement/proof or by removing the generalization, but it must be fixed before publication. The ambiguity in Section 2.3 should also be resolved by an explicit definition of the exterior-power extension."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this survey is worth having: it collects a genuinely scattered literature and gives several clean, correct proofs of the eigenvector-eigenvalue identity, so it will likely become the standard reference. Second, one of the advertised proofs is not correct. Section 2.3's coordinate-free Lemma 11 is true, but the proof as written relies on identity (22), which is false. In the notation of the paper, take T=diag(0,2,3) on C^3, v=e1, f=g=e2. Then the left side of (22) is zero, while the right side is nonzero (it equals ±3, depending on the Hodge star convention). The step 'if f is orthogonal to v then *f = v wedge eta and hence T(*f) vanishes' is wrong: T extends as a derivation, so T(v wedge eta)=v wedge T(eta), which is generally not zero. Lemma 11 itself is fine—it is essentially the identity in disguise—but it needs a different proof.\n\nWhat is actually new: not the main theorem. Jacobi 1834, Thompson 1966, and a dozen independent rediscoveries are documented honestly, with the authors explicitly refusing to guarantee completeness. The new material is the organization, several alternate proofs, and the mild generalization Proposition 17. The historical section is careful and does not oversell priority. The adjugate proof, Cramer-rule proof, perturbative proof, and Cauchy-Binet proof are all valid as far as I checked. The consistency checks in the introduction are useful, and there is even numerical code.\n\nSoft spots: the invalid coordinate-free proof is a real defect in a survey that promises multiple proofs, though it does not threaten the main theorem since the other proofs stand. Proposition 17 is attributed to a private communication and is relatively new; it looks right, but it has not been as thoroughly vetted as the rest. The historical completeness caveat is openly stated, so it is not a flaw. The citation pattern is appropriate; the paper leans on prior work rather than self-citations.\n\nWho is this for? Anyone who wants a citable source for the identity, a map of where it appeared, or a collection of derivations. It deserves a serious referee. My recommendation: send it out, but ask the authors to fix Section 2.3—either repair the argument or explicitly mark Lemma 11 as following from the adjugate proof. This is a corrigible issue, not a rejection.","headline":"A genuinely useful survey of a classical identity, with a real but localized flaw in one coordinate-free proof that does not threaten the main theorem.","tokens_in":25255,"tokens_out":4403,"would_cite":true,"duration_ms":42721,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["15A18","15A15","15A57"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves and surveys the eigenvector-eigenvalue identity, which expresses every eigenvector component magnitude of a Hermitian matrix as a ratio of eigenvalue-difference products.","keywords":["eigenvector-eigenvalue identity","Hermitian matrix","adjugate matrix","principal minor","characteristic polynomial","relative phase of eigenvectors","normal matrix","history of linear algebra"],"falsifier":"Compute a random Hermitian matrix with distinct eigenvalues in exact arithmetic and compare $|v_{i,j}|^2\\prod_{k\\ne i}(\\lambda_i(A)-\\lambda_k(A))$ with $\\prod_{k=1}^{n-1}(\\lambda_i(A)-\\lambda_k(M_j))$; a single mismatch would falsify Theorem 1. For the historical claim, a search of digitized pre-1834 mathematical literature for an equivalent cofactor or determinant identity would settle whether the survey's earliest-known date is right.","tokens_in":24345,"feed_emoji":"🧮","tokens_out":13984,"duration_ms":126180,"temperature":0.7,"pith_summary":"The paper establishes and surveys a basic identity: for any $n\\times n$ Hermitian matrix $A$, the squared magnitude of any component of a unit eigenvector is a ratio of two products of eigenvalue differences—one product over the eigenvalues of the minor formed by deleting that row and column, the other over the remaining eigenvalues of $A$. The identity is simple enough to prove from Cramer's rule or the adjugate matrix, yet the paper traces its appearance, in disguised or partial forms, across more than two dozen references spanning numerical linear algebra, graph theory, random matrix theory, inverse problems, and neutrino physics, with the earliest known precursor dated to 1834. A sympathetic reader should care because the identity converts a quantity usually obtained by full diagonalization—an eigenvector entry—into a direct spectral computation, and explains how such a basic fact could be independently rediscovered many times while remaining folklore in separate communities. The survey also shows how to recover the relative phases of eigenvector components and extends the identity to normal and diagonalizable matrices.","feed_headline":"Eigenvector entry sizes follow from eigenvalues alone","feed_subtitle":"One basic spectral formula fixes eigenvector component sizes; the survey traces two centuries of rediscovery.","key_machinery":"The load-bearing object is the adjugate matrix, the transpose of the matrix of signed cofactors. The key mechanism is the evaluation $$\\operatorname{adj}(\\lambda_i(A)I_n-A)=\\prod_{k\\ne i}(\\lambda_i(A)-\\lambda_k(A))\\,v_iv_i^*,$$ which shows that at an eigenvalue the adjugate of $\\lambda I_n-A$ collapses to a scalar multiple of the rank-one projection onto the corresponding eigenvector. Reading the $(j,j)$ entry of both sides turns the left side into the characteristic polynomial of the minor $M_j$ evaluated at $\\lambda_i(A)$, and the right side into the eigenvector component magnitude times the same product of eigenvalue differences; that equality is exactly the identity. This single adjugate identity carries the proof, its normal- and diagonalizable-matrix generalizations, and the off-diagonal relative-phase variant.","core_discovery":"On the paper's own terms, the central claim is Theorem 1. If $A$ is Hermitian with eigenvalues $\\lambda_1(A),\\dots,\\lambda_n(A)$, $v_i$ is a unit eigenvector for $\\lambda_i(A)$, and $M_j$ is the $(n-1)\\times(n-1)$ minor obtained by deleting row and column $j$, then $$|v_{i,j}|^2\\prod_{k\\ne i}(\\lambda_i(A)-\\lambda_k(A))=\\prod_{k=1}^{n-1}(\\lambda_i(A)-\\lambda_k(M_j)).$$ Using characteristic polynomials this reads $|v_{i,j}|^2p'_A(\\lambda_i(A))=p_{M_j}(\\lambda_i(A))$. The identity is not limited to Hermitian matrices: the same adjugate argument proves it for normal matrices, and a left-eigenvector variant holds for any diagonalizable matrix, including matrices over commutative rings. An off-diagonal analogue gives products $v_{i,j}v_{i,j'}$ and thereby the relative phases between components of one eigenvector. The survey presents several proofs (adjugate, Cramer's rule, coordinate-free, perturbative, Cauchy–Binet, and an alternate expression from random matrix theory) and a historical account in which the earliest known precursor appears in 1834 and the identity was independently rediscovered half a dozen times.","pith_inferences":["If the identity becomes part of standard linear algebra teaching, default derivations of eigenvector component magnitudes could shift from full diagonalization to this one-line spectral formula; the main bottleneck is that phases still require extra work, so the identity is not a full replacement for eigensolvers.","The rediscovery history suggests a testable sociological mechanism: a true but unnamed identity used only as a subsidiary tool, with no standard notation and no searchable keyword, can remain invisible across subfields even when repeatedly proved; a theorem-fingerprint database would be a direct remedy, and this survey gives a natural test entry.","The generalizations to matrices over commutative rings and to $J$-normal matrices point to a larger family of identities generated by compound-matrix manipulations; one could systematically enumerate all identities obtained by evaluating the $m$-th compound of $\\lambda I-A$ at an eigenvalue, a family that contains Proposition 17.","The infinite-dimensional extension is left open in the survey; if the resolvent-residue argument can be made rigorous with regularized determinants, the identity would supply spectral projection weights for Jacobi operators and other trace-class Hermitian perturbations."],"forward_implications":["For any Hermitian matrix, each squared eigenvector component can be computed from the eigenvalues of $A$ and of a single principal minor, without computing any eigenvector; checking the identity at one location requires only one characteristic-polynomial evaluation.","The relative phase between two components $v_{i,j}$ and $v_{i,k}$ of the same eigenvector can be recovered by applying the identity in two rotated bases and combining the resulting magnitudes, since individual eigenvector phases are arbitrary but relative phases are not.","The identity extends beyond Hermitian matrices: normal matrices satisfy the same formula unchanged, and diagonalizable matrices satisfy $\\det(\\lambda_i(A)I_{n-1}-M_j)=(\\prod_{k\\ne i}(\\lambda_i(A)-\\lambda_k(A)))\\,v_{i,j}w_{i,j}$, with $w_i$ a left eigenvector; this version survives over arbitrary commutative rings.","For structured matrices such as symmetric tridiagonal matrices, the identity gives closed-form eigenvector entries in terms of characteristic polynomials of contiguous minors, which underlies its uses in convergence analyses of eigensolvers and in inverse eigenvalue problems.","A residue version handles repeated eigenvalues: summing $|v_{i,j}|^2$ over the eigenspace of a common eigenvalue $\\lambda_*$ equals the residue of $p_{M_j}(\\lambda)/p_A(\\lambda)$ at $\\lambda_*$."],"supporting_citations":[{"why":"Supplies the earliest known precursor, an essentially equivalent cofactor identity for real symmetric quadratic forms.","marker":"[Jac1834]"},{"why":"Gives a rank-one perturbation identity that implies (2) as a limiting case and is proved by Cramer's rule with the Cauchy determinant.","marker":"[Löw1934]"},{"why":"First appearance the survey knows of in essentially the present form, including normal matrices and repeated eigenvalues.","marker":"[Tho1966]"},{"why":"Reproduces the tridiagonal version and is the source of the adjugate proof used in Section 2.1.","marker":"[Par1980]"},{"why":"Cramer's rule is the core of the resolvent proof: the $jj$ entry of $(\\lambda I-A)^{-1}$ is a ratio of minor to full characteristic polynomials.","marker":"[Cra1750]"},{"why":"Provides the Cauchy determinant identity used in one derivation of (2).","marker":"[Cau1841]"},{"why":"The authors' original preprint that introduced the identity to the current literature and supplied the adjugate and Cauchy-Binet proofs.","marker":"[DPTZ2019]"},{"why":"The popular account whose publicity triggered the crowdsourced collection of previous appearances, load-bearing for the historical-dissemination claim.","marker":"[Wol2019]"}],"fun_headline_variants":["Eigenvector entries? Eigenvalues alone determine their sizes","Two centuries old: identity links eigenvectors to eigenvalues","Eigenvector components sized by eigenvalues via simple formula","One identity: eigenvector sizes from eigenvalues, rediscovered many times","From eigenvalues to eigenvector entries: a 200-year-old formula"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption for the survey's historical narrative is that the crowdsourced reference list is complete; the authors explicitly say they cannot guarantee that an even earlier form of the identity has not appeared, so a newly found earlier precursor would overturn the 1834 'earliest known' claim while leaving the mathematical identity untouched.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvector entries? Eigenvalues alone determine their sizes","Two centuries old: identity links eigenvectors to eigenvalues","Eigenvector components sized by eigenvalues via simple formula","One identity: eigenvector sizes from eigenvalues, rediscovered many times","From eigenvalues to eigenvector entries: a 200-year-old formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000758,"raw_usage":{"total_tokens":3446,"prompt_tokens":1102,"completion_tokens":2344,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":718,"completion_tokens_details":{"reasoning_tokens":2262}},"tokens_in":718,"tokens_out":2344,"duration_ms":18584,"temperature":1.0,"reasoning_tokens":2262,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:00:56.713772+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute a random Hermitian matrix with distinct eigenvalues in exact arithmetic and compare $|v_{i,j}|^2\\prod_{k\\ne i}(\\lambda_i(A)-\\lambda_k(A))$ with $\\prod_{k=1}^{n-1}(\\lambda_i(A)-\\lambda_k(M_j))$; a single mismatch would falsify Theorem 1. For the historical claim, a search of digitized pre-1834 mathematical literature for an equivalent cofactor or determinant identity would settle whether the survey's earliest-known date is right.","supporting_citations":[],"review_version":1}