Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra

Peter B. Denton; Stephen J. Parke; Terence Tao; Xining Zhang

REVIEW 2 major objections 5 minor 3 cited by

Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 1908.03795 v4 pith:W5N3R3RX submitted 2019-08-10 math.RA hep-phhep-thmath-phmath.MP

Peter B. Denton , Stephen J. Parke , Terence Tao , Xining Zhang This is my paper · ORCID

classification math.RAhep-phhep-thmath-phmath.MP MSC 15A1815A1515A57

keywords eigenvector-eigenvalueidentityHermitianmatrixadjugateprincipalminorcharacteristicpolynomialrelativephaseofeigenvectorsnormalhistorylinearalgebra

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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 $\lambda_*$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Who 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.

Referee Report

2 major / 5 minor

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."

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 (2)

[Section 2.7, Proposition 17 and Eq. (24)] 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 2.3, Lemma 11 and Eq. (22)] 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.

minor comments (5)

[Section 2.3, after Lemma 11] 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 2.7, proof of Proposition 17] 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 1, consistency check (x)] 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 2.2, paragraph after Eq. (18)] 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 2.2, before Eq. (17)] The phrase "An alternate approach way to arrive at (2) from (14)" should be "An alternate way to arrive at (2) from (14)".

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the identity is proven from independent standard linear-algebra facts, and self-citations are historical rather than load-bearing.

full rationale

Theorem 1 is not assumed anywhere in its own proofs. Section 2.1 derives it from adjugate/Cramer's rule; Section 2.2 from the resolvent and residue extraction; Section 2.4 from a first-order perturbation expansion; Section 2.5 from a Cauchy-Binet-type lemma proved independently by determinant expansion; Section 2.6 from Schur complementation and Lemma 15, which is proved directly from the eigenvector equation. In each case the target identity (2)/(4) appears only as the conclusion, not as an input. The references to the authors' own preprint [DPTZ2019] and paper [DPZ2020] attribute rediscovery and reproduce the arguments; they do not carry the derivational weight. The historical priority claim is explicitly qualified ('we are not able to guarantee that there is not an even earlier place in the literature where some form of this identity has appeared'), which is a caveat about completeness rather than a circular justification. The possible invalidity of the coordinate-free proof in Section 2.3 (the assertion that T(*f) vanishes when *f = v ∧ η is false) concerns correctness of one proof, not circularity, and the identity remains independently established by the other proofs.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central identity is proven from standard background results; no ad hoc assumptions or free parameters are introduced. The historical survey relies on the acknowledged incompleteness of the reference list (Section 5), which does not affect the mathematics.

assumptions (6)

standard math Spectral theorem for Hermitian matrices
Used to diagonalize A and express adj(A) in the adjugate proof (Section 2.1).
standard math Cramer's rule and adjugate identity adj(A)A = det(A)I
Basis of the adjugate proof and Cramer's rule proof (Sections 2.1, 2.2).
standard math Cauchy interlacing inequalities
Used in consistency checks and to relate eigenvalues of A and M_j (Introduction).
standard math Cauchy determinant identity
Used in the Cramer's rule proof via the linear system (17)-(18) (Section 2.2).
standard math Schur complement determinant formula
Used in the alternate expression for eigenvector components (Section 2.6).
standard math Limiting argument for repeated eigenvalues
Several proofs first assume simple spectrum then extend by continuity (Sections 2.2, 2.4).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra." pith.science (2026). https://pith.science/paper/W5N3R3RX

@misc{pith2026190803795,
  author       = {Pith},
  title        = {Pith review of: Eigenvectors from eigenvalues: A survey of a basic identity in linear algebra},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W5N3R3RX}},
  note         = {Machine review of arXiv:1908.03795}
}

abstract

If $A$ is an $n \times n$ Hermitian matrix with eigenvalues $\lambda_1(A),\dots,\lambda_n(A)$ and $i,j = 1,\dots,n$, then the $j^{\mathrm{th}}$ component $v_{i,j}$ of a unit eigenvector $v_i$ associated to the eigenvalue $\lambda_i(A)$ is related to the eigenvalues $\lambda_1(M_j),\dots,\lambda_{n-1}(M_j)$ of the minor $M_j$ of $A$ formed by removing the $j^{\mathrm{th}}$ row and column by the formula $$ |v_{i,j}|^2\prod_{k=1;k\neq i}^{n}\left(\lambda_i(A)-\lambda_k(A)\right)=\prod_{k=1}^{n-1}\left(\lambda_i(A)-\lambda_k(M_j)\right)\,.$$ We refer to this identity as the \emph{eigenvector-eigenvalue identity} and show how this identity can also be used to extract the relative phases between the components of any given eigenvector. Despite the simple nature of this identity and the extremely mature state of development of linear algebra, this identity was not widely known until very recently. In this survey we describe the many times that this identity, or variants thereof, have been discovered and rediscovered in the literature (with the earliest precursor we know of appearing in 1834). We also provide a number of proofs and generalizations of the identity.

Figures

Figures reproduced from arXiv: 1908.03795 by the authors.

**Figure 1.** The citation graph of all the references in the literature we are aware of (predating the current survey) that mention some variant of the eigenvector-eigenvalue identity. To reduce clutter, transitive references (e.g., a citation of a paper already cited by another paper in the bibliography) are omitted. Note the very weakly connected nature of the graph, with many early initial references not being (transitively)… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Block Lanczos algorithm for lattice QCD spectroscopy and matrix elements
hep-lat 2024-12 conditional novelty 7.0 of 10

Block Lanczos is shown to generalize the GEVP method for lattice QCD correlator matrices, providing faster convergence, two-sided error bounds, and improved signal-to-noise.
Fibonacci Fast Convergence for Neutrino Oscillations in Matter
hep-ph 2019-09 conditional novelty 7.0 of 10

Applying the vacuum 2-3 rotation and then repeatedly eliminating the largest off-diagonal matrix element makes the effective Hamiltonian converge with an error order that grows like the Fibonacci sequence.
Perturbative determination of CP phase in CKM matrix using rephasing invariants of hierarchical mass matrices and their inverses
hep-ph 2026-07 conditional novelty 6.0 of 10

The CKM CP phase is expressed as a fourth-order rephasing invariant constructed from the down-quark mass matrix and its inverse via perturbative singular value decomposition.

Reference graph

Works this paper leans on

59 extracted references · 56 canonical work pages · cited by 3 Pith papers

[1]

Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics Series . For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964

1964
[2]

Baryshnikov

Yu. Baryshnikov. G UE s and queues. Probab. Theory Related Fields , 119(2):256--274, 2001

work page 2001
[3]

Bebiano, S

N. Bebiano, S. Furtado, and J. da Provid\^ e ncia. On the eigenvalues of principal submatrices of J -normal matrices. Linear Algebra Appl. , 435(12):3101--3114, 2011

work page 2011
[4]

Boley and G

D. Boley and G. H. Golub. Inverse eigenvalue problems for band matrices. In Numerical analysis ( P roc. 7th B iennial C onf., U niv. D undee, D undee, 1977) , pages 23--31. Lecture Notes in Math., Vol. 630, 1978

work page 1977
[5]

An explicit formula for the eigenvectors of acyclic matrices and weighted trees, 2016

Asghar Bahmani and Dariush Kiani. An explicit formula for the eigenvectors of acyclic matrices and weighted trees, 2016

work page 2016
[6]

A S urvey of M odern A lgebra

Garrett Birkhoff and Saunders MacLane. A S urvey of M odern A lgebra . Macmillan Company, New York, 1941

work page 1941
[7]

Billey and Bridget E

Sara C. Billey and Bridget E. Tenner. Fingerprint databases for theorems. Notices Amer. Math. Soc. , 60(8):1034--1039, 2013

work page 2013
[8]

Exercices d'analyse et de physique math\'ematique

Augustin Louis Cauchy. Exercices d'analyse et de physique math\'ematique. vol. 2. Bachelier , page 154, 1841

work page

Show all 59 references

[9]

Chu and Gene H

Moody T. Chu and Gene H. Golub. Structured inverse eigenvalue problems. Acta Numer. , 11:1--71, 2002

2002
[10]

Note on eigenvectors from eigenvalues, 2019

Xiaomei Chen. Note on eigenvectors from eigenvalues, 2019

2019
[11]

Richard W. Cottle. Manifestations of the S chur complement. Linear Algebra Appl. , 8:189--211, 1974

1974
[12]

Introduction \`a l'analyse des lignes courbes alg\'ebriques

Gabriel Cramer. Introduction \`a l'analyse des lignes courbes alg\'ebriques. Geneva , pages 657--659, 1750
[13]

Cvetkovi\'c, P

D. Cvetkovi\'c, P. Rowlinson, and S.K. Simi\'c. Star complements and exceptional graphs. Linear Algebra and its Applications , 423(1):146 -- 154, 2007. Special Issue devoted to papers presented at the Aveiro Workshop on Graph Spectra

2007
[14]

Matrix models for beta ensembles

Ioana Dumitriu and Alan Edelman. Matrix models for beta ensembles. J. Math. Phys. , 43(11):5830--5847, 2002

2002
[15]

James W. Demmel. Applied numerical linear algebra . Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997

1997
[16]

Peter B. Denton. Eigenvalue-eigenvector identity code. https://github.com/PeterDenton/Eigenvector-Eigenvalue-Identity, 2019

2019
[17]

On C auchy's inequalities for H ermitian matrices

Emeric Deutsch and Harry Hochstadt. On C auchy's inequalities for H ermitian matrices. Amer. Math. Monthly , 85(6):486--487, 1978

1978
[18]

Deift, L

P. Deift, L. C. Li, T. Nanda, and C. Tomei. The T oda flow on a generic orbit is integrable. Comm. Pure Appl. Math. , 39(2):183--232, 1986

1986
[19]

Denton , Stephen J

Peter B. Denton , Stephen J. Parke , Terence Tao , and Xining Zhang . Eigenvectors from Eigenvalues . arXiv e-prints , page arXiv:1908.03795v1, Aug 2019

1908 arXiv
[20]

Denton, Stephen J

Peter B. Denton, Stephen J. Parke, and Xining Zhang. Neutrino oscillations in matter via eigenvalues . Phys. Rev. D , 101(9):093001, 2020

2020
[21]

Semicircle law on short scales and delocalization of eigenvectors for W igner random matrices

L\' a szl\' o Erd o s, Benjamin Schlein, and Horng-Tzer Yau. Semicircle law on short scales and delocalization of eigenvectors for W igner random matrices. Ann. Probab. , 37(3):815--852, 2009

2009
[22]

Forrester and Jiyuan Zhang

Peter J. Forrester and Jiyuan Zhang . Co-rank 1 projections and the randomised Horn problem . arXiv e-prints , page arXiv:1905.05314, May 2019

1905 arXiv
[23]

F. R. Gantmacher. The theory of matrices. Vol I . Translated by K. A. Hirsch. Chelsea Publishing Co., New York, 1959

1959
[24]

Eisenstat

Ming Gu and Stanley C. Eisenstat. A stable and efficient algorithm for the rank-one modification of the symmetric eigenproblem. SIAM J. Matrix Anal. Appl. , 15(4):1266--1276, 1994

1994
[25]

State transfer in strongly regular graphs with an edge perturbation, 2017

Chris Godsil, Krystal Guo, Mark Kempton, and Gabor Lippner. State transfer in strongly regular graphs with an edge perturbation, 2017

2017
[26]

The neutrino-neutrino interaction effects in supernovae: the point of view from the matter basis

S\'ebastien Galais, James Kneller, and Cristina Volpe. The neutrino-neutrino interaction effects in supernovae: the point of view from the matter basis . J. Phys. , G39:035201, 2012

2012
[27]

Graham M. L. Gladwell. Inverse problems in vibration , volume 119 of Solid Mechanics and its Applications . Kluwer Academic Publishers, Dordrecht, second edition, 2004

2004
[28]

C. D. Godsil and B. D. McKay. Spectral conditions for the reconstructibility of a graph. J. Combin. Theory Ser. B , 30(3):285--289, 1981

1981
[29]

C. D. Godsil. Algebraic combinatorics . Chapman and Hall Mathematics Series. Chapman & Hall, New York, 1993

1993
[30]

When can perfect state transfer occur? Electron

Chris Godsil. When can perfect state transfer occur? Electron. J. Linear Algebra , 23:877--890, 2012

2012
[31]

Gene H. Golub. Some modified matrix eigenvalue problems. SIAM Rev. , 15:318--334, 1973

1973
[32]

Gaveau and L

B. Gaveau and L. S. Schulman. Limited quantum decay. J. Phys. A , 28(24):7359--7374, 1995

1995
[33]

Golub and Charles F

Gene H. Golub and Charles F. Van Loan. Matrix computations , volume 3 of Johns Hopkins Series in the Mathematical Sciences . Johns Hopkins University Press, Baltimore, MD, 1983

1983
[34]

Elias M. Hagos. Some results on graph spectra. Linear Algebra and its Applications , 356(1):103 -- 111, 2002

2002
[35]

Paul R. Halmos. Finite D imensional V ector S paces . Annals of Mathematics Studies, no. 7. Princeton University Press, Princeton, N.J., 1942

1942
[36]

C. G. J. Jacobi. De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares in alias binas tranformandis, quae solis quadratis variabilium constant; una cum variis theorematis de tranformatione etdeterminatione integralium multiplicium. J. Reine A...
[37]

Normalized modes at selected points without normalization

Eduardo Kausel. Normalized modes at selected points without normalization. Journal of Sound and Vibration , 420:261--268, April 2018

2018
[38]

Computation of eigenvalues and eigenvectors for mesh problems: algorithms and error estimates (in Russian)

Andrew Knyazev. Computation of eigenvalues and eigenvectors for mesh problems: algorithms and error estimates (in Russian) . PhD thesis, Department of Numerical Mathematics, USSR Academy of sciences, Moscow, 1986

1986
[39]

A. V. Knyazev and A. L. Skorokhodov. On exact estimates of the convergence rate of the steepest ascent method in the symmetric eigenvalue problem. Linear Algebra Appl. , 154/156:245--257, 1991

1991
[40]

On the largest eigenvalue of a graph

Qiao Li and Ke Qin Feng. On the largest eigenvalue of a graph. Acta Math. Appl. Sinica , 2(2):167--175, 1979

1979
[41]

o wner. \

Karl L \" o wner. \" U ber monotone M atrixfunktionen. Math. Z. , 38(1):177--216, 1934

1934
[42]

Mukherjee and Kali Kinkar Datta

Asok K. Mukherjee and Kali Kinkar Datta. Two new graph-theoretical methods for generation of eigenvectors of chemical graphs. Proceedings of the Indian Academy of Sciences - Chemical Sciences , 101(6):499--517, Dec 1989

1989
[43]

On the eigenvalues of principal submatrices of normal, H ermitian and symmetric matrices

Peter Nylen, Tin Yau Tam, and Frank Uhlig. On the eigenvalues of principal submatrices of normal, H ermitian and symmetric matrices. Linear and Multilinear Algebra , 36(1):69--78, 1993

1993
[44]

The computation of eigenvalues and eigenvectors of very large sparse matrices

Christopher Conway Paige. The computation of eigenvalues and eigenvectors of very large sparse matrices . PhD thesis, London University Institute of Computer Science, 1971

1971
[45]

Beresford N. Parlett. The symmetric eigenvalue problem . Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics

1980
[46]

More on the eigenvectors-from-eigenvalues identity, 2019

Malgorzata Stawiska. More on the eigenvectors-from-eigenvalues identity, 2019

2019
[47]

R. C. Thompson. Principal submatrices of normal and H ermitian matrices. Illinois J. Math. , 10:296--308, 1966

1966
[48]

R. C. Thompson. Principal submatrices. IV . O n the independence of the eigenvalues of different principal submatrices. Linear Algebra and Appl. , 2:355--374, 1969

1969
[49]

Thompson and P

R.C. Thompson and P. McEnteggert. Principal submatrices ii: the upper and lower quadratic inequalities. Linear Algebra and its Applications , 1:211--243, 1968

1968
[50]

S. Toshev. On T violation in matter neutrino oscillations . Mod. Phys. Lett. , A6:455--460, 1991

1991
[51]

Random matrices: Universality of local eigenvalue statistics

Terence Tao and Van Vu. Random matrices: Universality of local eigenvalue statistics. Acta Math. , 206(1):127--204, 2011

2011
[52]

Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks

Piet Van Mieghem . Graph eigenvectors, fundamental weights and centrality metrics for nodes in networks . arXiv e-prints , page arXiv:1401.4580, Jan 2014

2014 arXiv
[53]

Graph spectra for complex networks

Piet Van Mieghem. Graph spectra for complex networks . Cambridge University Press, Cambridge, 2011

2011
[54]

G. E. S ilov. Matematicheski i analiz. Konechnomernye line i nye prostranstva . Izdat. ``Nauka'', Moscow, 1969

1969
[55]

H. F. Weinberger. Error bounds in the R ayleigh- R itz approximation of eigenvectors. J. Res. Nat. Bur. Standards Sect. B , 64B:217--225, 1960

1960
[56]

J. H. Wilkinson. Rounding errors in algebraic processes . Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963

1963
[57]

Neutrinos lead to unexpected discovery in basic math

Natalie Wolchover. Neutrinos lead to unexpected discovery in basic math. Quanta Magazine , Nov 2019

2019
[58]

Theories and Methods of Matrix Calculations (in Chinese)

Shufang Xu. Theories and Methods of Matrix Calculations (in Chinese) . Peking University Press, Beijing, 1995

1995
[59]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence '...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Milton Abramowitz and Irene A. Stegun. Handbook of mathematical functions with formulas, graphs, and mathematical tables , volume 55 of National Bureau of Standards Applied Mathematics Series . For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964

1964

[2] [2]

Baryshnikov

Yu. Baryshnikov. G UE s and queues. Probab. Theory Related Fields , 119(2):256--274, 2001

work page 2001

[3] [3]

Bebiano, S

N. Bebiano, S. Furtado, and J. da Provid\^ e ncia. On the eigenvalues of principal submatrices of J -normal matrices. Linear Algebra Appl. , 435(12):3101--3114, 2011

work page 2011

[4] [4]

Boley and G

D. Boley and G. H. Golub. Inverse eigenvalue problems for band matrices. In Numerical analysis ( P roc. 7th B iennial C onf., U niv. D undee, D undee, 1977) , pages 23--31. Lecture Notes in Math., Vol. 630, 1978

work page 1977

[5] [5]

An explicit formula for the eigenvectors of acyclic matrices and weighted trees, 2016

Asghar Bahmani and Dariush Kiani. An explicit formula for the eigenvectors of acyclic matrices and weighted trees, 2016

work page 2016

[6] [6]

A S urvey of M odern A lgebra

Garrett Birkhoff and Saunders MacLane. A S urvey of M odern A lgebra . Macmillan Company, New York, 1941

work page 1941

[7] [7]

Billey and Bridget E

Sara C. Billey and Bridget E. Tenner. Fingerprint databases for theorems. Notices Amer. Math. Soc. , 60(8):1034--1039, 2013

work page 2013

[8] [8]

Exercices d'analyse et de physique math\'ematique

Augustin Louis Cauchy. Exercices d'analyse et de physique math\'ematique. vol. 2. Bachelier , page 154, 1841

work page