Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices I: Theory

A. T. Walden; J. F. Lutzeyer

REVIEW 2 major objections 2 minor 28 references

Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices I: Theory

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

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Pith's one-line read Polynomial transform extends Davis–Kahan eigenvector bound to any block

desk verdict The polynomial-transform extension of Davis-Kahan is a genuinely new conditional bound, but the advertised universality for arbitrary spectra is not proven; Theorem 3's between-eigengap formula is a nice standalone contribution. read the letter →

arxiv 1908.03462 v1 pith:PYUEPTYX submitted 2019-08-09 math.ST stat.TH

J. F. Lutzeyer , A. T. Walden This is my paper · ORCID

classification math.STstat.TH MSC 15A42

keywords Davis-Kahantheorempolynomialmatrixtransformaffineeigenspacedistancegraphshiftoperatorcanonicalangleseigenvectorperturbation

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

This paper extends the Davis–Kahan theorem, the standard tool for bounding how far the eigenspaces of two symmetric matrices can be, so that it applies to any corresponding consecutive block of eigenvectors rather than only the leading r. The device is a polynomial matrix transform: applying p to one matrix leaves its eigenvectors unchanged but moves its eigenvalues, so the separation required by Davis–Kahan is shifted from the fixed spectra to transformation parameters the user controls. The paper proves that if the transform satisfies two explicit constraints on where the transformed eigenvalues lie, the usual subspace-distance bound holds with denominator δ_i and numerator ||p(Φ)-Ψ||, and the standard theorem is a special case. Because the transform can be chosen to shrink the matrix difference and enlarge the eigengap, the new bounds are guaranteed to be no worse than classical Davis–Kahan bounds. A proof-of-concept on d-regular graphs shows the method automatically recovers the exact eigenvector agreement between unnormalised and normalised graph Laplacians.

What carries the argument

The central object is the polynomial matrix transform p(Φ): by the spectral mapping property for symmetric matrices, if (φ, w) is an eigenpair of Φ then (p(φ), w) is an eigenpair of p(Φ), so eigenvectors are untouched while eigenvalues are relocated. The paper's Constraints 1 and 2A/2B are the load-bearing conditions: they ensure that after transformation, the r eigenvalues of p(Φ) in the block of interest and the corresponding block of Ψ can be enclosed in Davis–Kahan intervals S1, S2 separated by δ_i > 0. With these constraints, the paper obtains the bound ||W_j W_j^T (I - V_j V_j^T)|| ≤ ||p(Φ) - Ψ|| / δ_i for every unitarily invariant norm, and the Frobenius variant with constant c_{n,r} = $\sqrt$(2 min(r, n - r)) from Lemma 2, which relates the two subspace distance metrics via canonical angles.

What would settle it

A concrete negative test is to enumerate all possible orderings of n transformed eigenvalues achievable by polynomials of degree at most d for a given spectrum, and check whether any ordering places the r middle eigenvalues between the outside ones as Constraints 1 require; a spectrum where no admissible ordering exists, despite nonzero boundary eigengaps, would falsify the claim that the extension applies universally to any two symmetric matrices.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 5 and Corollary 1: for symmetric matrices Φ and Ψ with nonzero jth and (j+r)th eigengaps, any polynomial p satisfying Constraints 1 and 2A (or 2B) yields the bound ||W_j W_j^T (I - V_j V_j^T)|| ≤ ||p(Φ) - Ψ|| / δ_i for every unitarily invariant norm, and ||W_j - V_j Q||_F ≤ c_{n,r} ||p(Φ) - Ψ||_2 / δ_i for some orthogonal Q. The paper asserts that this extension applies to any two symmetric matrices, excluding degenerate cases, and to any r consecutive corresponding eigenvectors, with the original Davis–Kahan theorem recovered as the special case j = 0, p(x) = x; consequently the extended bounds are at least as tight as classical ones. The transform is the mechanism: since eigenvectors of p(Φ) are eigenvectors of Φ, the subspace distance is unchanged while the spectral positioning is adjustable. The proof-of-concept example with d-regular graphs is central: for these graphs L_sym = $d^{{-1}}$L, so the affine transform with c1 = $d^{{-1}}$, c0 = 0 maps L exactly to L_sym and the extended bound is identically zero, whereas the standard Davis–Kahan bound is generally nonzero.

Load-bearing premise

The theorem's breadth rests on the assumption that for any two symmetric spectra with nonzero jth and (j+r)th eigengaps, some polynomial (at least an affine one) exists whose transformed eigenvalues satisfy Constraints 1 and 2; this existence is stated and demonstrated in examples but not proven here.

Editorial extensions

If this is right

For any two symmetric matrices whose spectra have nonzero boundary eigengaps around the compared block, the extended bound applies even when the compared eigenvalues occupy opposite ends of the spectrum, a case the classical theorem cannot handle.
Because the standard Davis–Kahan theorem is the special case j = 0 with p(x) = x, the new bound is at least as tight as the classical bound in every setting where both apply.
Users can choose the polynomial to minimize ||p(Φ) - Ψ||_2 and maximize the interval separation δ_i, making the bound an optimization target rather than a fixed quantity.
In the d-regular graph example, the bound is identically zero when the transform maps the unnormalised Laplacian to the normalised one, so the method automatically identifies the equality of the corresponding eigenspaces.

Reading between the lines

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

The authors leave implicit that the zero-bound phenomenon provides a practical certificate: if a user can find an admissible transform mapping Φ exactly to Ψ, the bound certifies exact eigenspace agreement; applying this to empirically estimated graph spectra would give a testable equivalence test between graph shift operators.
An extension not pursued here is to apply the same transform device to normal matrices or to non-symmetric spectra via a Jordan-form version of the spectral mapping, though the interval geometry would need substantial reworking.
The dependence of the bound on δ_i suggests a data-driven heuristic the paper does not develop: among candidate ranks r and offsets j, choose the block that maximizes the between-matrix separation, since a larger denominator directly shrinks the bound before any optimization of p.

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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

Here's my read. The paper extends Davis-Kahan by applying a polynomial matrix transform to one of the two matrices. Theorem 5 is the main new result: if a polynomial p satisfies their Constraints 1 and 2, then you get the standard DK inequality with p(Phi) - Psi in the numerator and a transformed eigengap in the denominator. I checked the proof in Appendix D and it is correct; the constraints are exactly what is needed for the interval requirements of the original theorem. Corollary 1 follows cleanly via their Lemma 2. Theorem 3, the between-matrix eigengap bound for the first r eigenvectors, is a nice self-contained contribution; I had not seen the max(phi_{r+1}-psi_r, psi_{r+1}-phi_r) denominator before.

The core idea is genuinely useful. Transforming one matrix preserves eigenvectors but moves eigenvalues, so you can create a positive separation where the standard DK setup has none. That matters for comparing adjacency versus Laplacian eigenvectors and for tightening perturbation bounds.

The soft spot is the gap between the abstract's 'any two symmetric matrices' and what is actually proved. Theorem 5 is conditional on the existence of a polynomial satisfying Constraints 1 and 2. The paper never proves such a polynomial always exists, even among affine transforms. Section VI gives formulas for affine maps but no feasibility argument. The proof of concept uses d-regular graphs, where Lsym = d^{-1}L exactly, so the affine transform is known by inspection and gives a zero bound. That is a good illustration, not a demonstration of the general claim. The missing existence argument is the difference between 'we can tune a bound when a transform exists' and 'we can always find such a transform.' If the authors want the universal claim, they need to prove it, or at least prove it for affine transforms, or soften the language to 'whenever an admissible transform exists.'

The paper is clearly written, the conditional result is solid, and the related literature is handled fairly. The typos in Appendix D ('Contraints' and a mix-up of 2A/2B) are minor.

My verdict: this deserves a serious referee. It is not ready as is, but the conditional theorem and Theorem 3 are publishable contributions, and the overclaim is fixable. I would not desk-reject.

Referee Report

2 major / 2 minor

Summary. The paper extends the Davis-Kahan (DK) theorem to compare the subspaces spanned by any r consecutive eigenvectors of two symmetric matrices. The main idea is to apply a polynomial matrix transform p(·) to one of the matrices, which preserves eigenvectors but moves eigenvalues so that a DK-type bound becomes applicable. The central result, Theorem 5 and Corollary 1, states that if a polynomial p satisfies Constraints 1 and 2A or 2B, then the projector distance and the Frobenius distance (up to an orthogonal rotation) are bounded by ||p(Φ)-Ψ|| divided by an eigenvalue-separation factor δ. Theorem 3 treats the special case of the first r eigenvectors and proves that, under nonzero r-th eigengaps, a DK interval always exists, yielding a bound in terms of a between-matrix eigengap. A proof-of-concept example for d-regular graph Laplacians is given, where the affine transform c1 = d^{-1}, c0 = 0 exactly maps L to Lsym, and the algorithm from a companion paper is said to identify this transform automatically.

Significance. The paper offers a genuinely useful idea: using polynomial matrix transforms to relax the restrictive eigenvalue-interval structure of the DK theorem and to tighten the resulting bound. The conditional Theorems 5 and Corollary 1 are carefully proved in Appendix D, and Lemma 2 provides a sharp constant c_{n,r} = sqrt(2 min(r,n-r)) for the relation between the two distance metrics. Theorem 3 is a clean and valid extension of the DK theorem to first-r eigenvector comparisons. If the existence of admissible transforms for arbitrary spectra is established, this work would have broad applicability in graph signal processing, spectral clustering, and spiked covariance models. However, the advertised universality for 'any two symmetric matrices' is not proved in the manuscript, and the proof of concept depends on a special spectral relation and on an algorithm deferred to a companion paper.

major comments (2)

[Section II (last paragraph), Section V-C/V-D, Section VII]
[Section V-D (Remark 4 and the paragraph after Corollary 1)]

minor comments (2)

[Appendix D (proof of Theorem 5, interval choice (8))]
[Throughout]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is a conditional upper bound whose proof invokes only the classical Davis–Kahan theorem, and the proof-of-concept uses the independent identity L_sym = d^{-1}L for d-regular graphs.

full rationale

The derivation chain is self-contained and does not reduce to its inputs. Theorem 5 and Corollary 1 state an upper bound under explicit sufficient conditions on a polynomial p (Constraints 1 and 2A or 2B), and the proof checks that these conditions make the classical Davis–Kahan theorem applicable to the pair (p(Phi), Psi); this is a legitimate conditional proof, not a fitting procedure. The polynomial transform is not defined in terms of the target distance; it is chosen to manipulate eigenvalues while preserving eigenvectors, so the bound remains a genuine upper bound for every admissible p. Optimizing p over that bound is an optimization of an upper bound, not a claim that the bound is attained or a renamed fit. Theorem 3 is also proved directly from the Davis–Kahan theorem with an explicit interval construction. The proof-of-concept example compares eigenvectors of L and Lsym for d-regular graphs using the independently known exact relation Lsym = d^{-1}L; this is a sanity check, not a prediction fitted to the data, and it does not smuggle in the conclusion. The paper does claim, in the Introduction, that for any two symmetric matrices with nonzero boundary eigengaps a valid DK interval setup can always be found, but the existence of a polynomial satisfying the constraints is asserted rather than proved in this manuscript; that is a missing proof or an overstatement of universality, but not circularity. References to the companion paper (Paper II) provide the computational algorithm used in the example, but the theorem itself does not depend on that algorithm, so the self-citation is not load-bearing. No step in the derivation is equivalent by construction to its own conclusion, and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The derivation builds on the classical Davis-Kahan theorem and standard spectral facts. The only genuinely extra burden is the asserted but unproven existence of a transformation satisfying the interval constraints, which the universal applicability claim depends on. No new physical or mathematical entities are postulated.

free parameters (1)

affine transform parameters c1, c0 (general polynomial coefficients in the extended method) = c1 = 1/d, c0 = 0 in the d-regular graph example
These are user-chosen transformation parameters that must satisfy Constraints 1 and 2 and are then optimized to minimize the bound. They are not fitted to data and do not enter the theorem as unknowns, but the proof-of-concept example fixes them to the known scaling between Lsym and L.

assumptions (4)

standard math Davis-Kahan theorem as stated in Theorem 2
The entire extension builds on this external theorem; it is taken as given.
domain assumption Symmetric matrices over R with ordered eigenvalues and orthonormal eigenbases
The paper restricts to symmetric matrices Phi, Psi in R^{n x n} throughout.
domain assumption Assumption 2: nonzero jth and (j+r)th eigengaps in both spectra
Required for unambiguous contiguous eigenvector blocks and valid DK intervals; stated as Assumption 2.
ad hoc to paper Existence of a polynomial p satisfying Constraints 1 and 2 for arbitrary spectra
Theorem 5 is conditional on this existence; the paper asserts it is always achievable but defers the constructive algorithm to Paper II, so it functions as an unproved premise underlying the universal applicability claim.

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Pith. "Pith review of Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices I: Theory." pith.science (2026). https://pith.science/paper/PYUEPTYX

@misc{pith2026190803462,
  author       = {Pith},
  title        = {Pith review of: Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices I: Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PYUEPTYX}},
  note         = {Machine review of arXiv:1908.03462}
}

abstract

The Davis-Kahan theorem can be used to bound the distance of the spaces spanned by the first $r$ eigenvectors of any two symmetric matrices. We extend the Davis-Kahan theorem to apply to the comparison of the union of eigenspaces of any two symmetric matrices by making use of polynomial matrix transforms and in so doing, tighten the bound. The transform allows us to move requirements present in the original Davis-Kahan theorem, from the eigenvalues of the compared matrices on to the transformation parameters, with the latter being under our control. We provide a proof of concept example, comparing the spaces spanned by the unnormalised and normalised graph Laplacian eigenvectors for $d$-regular graphs, in which the correct transform is automatically identified.

Figures

Figures reproduced from arXiv: 1908.03462 by the authors.

**Figure 2.** A possible eigenvalue configuration used to illustr [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Dots represent attained distances, ρ1(W0, V0), diamonds represent bound values calculated from Corollary 1, using an affine matrix transformation, and ‘x’ symbols represent bound values calculated from Theorem 3. In the above example both the standard DK theorem and our extended applicability version were able to be applied in comparing L and Lsym. In other cases, such as when comparing A and L, the standard DK the… view at source ↗

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Pith tools

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

[1] [1]

D. S. Bernstein, Matrix Mathematics: Theory, Facts, and F ormulas (2nd. Edition), Princeton, NJ: Princeton University Press, 2009

work page 2009

[2] [2]

Bhatia, Matrix Analysis , Berlin, Germany: Springer Science & Business Media, 2013

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work page 2013

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work page 2019

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C. Davis & W. M. Kahan, “The Rotation of Eigenvectors by a P er- turbation. III,” SIAM Journal on Numerical Analysis , vol. 7, pp. 1–46, 1970

work page 1970

[5] [5]

Spectral Projector-Based Gra ph Fourier Transforms,

J. A. Deri & J. M. F. Moura, “Spectral Projector-Based Gra ph Fourier Transforms,” IEEE Journal of Selected Topics in Signal Processing , vol. 11, pp. 785–795, 2017

work page 2017

[6] [6]

Learning Gra phs from Data: A signal representation perspective,

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[7] [7]

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A. Edelman, T. A. Arias & S. T. Smith, “The geometry of algo rithms with orthogonality constraints,” SIAM Journal on Matrix Analysis and Applications, vol. 20, pp. 303–353, 1998

work page 1998

[8] [8]

Unperturbed: spectral analysis beyond Davis–Kahan,

J. Eldridge, M. Belkin & Y . Wang, “Unperturbed: spectral analysis beyond Davis–Kahan,” Proceedings of Algorithmic Learning Theory , vol. 83, pp. 321–358, 2018

work page 2018