Complex Frequency as Generalized Eigenvalue

Federico Milano; Nikolas Sofos

arxiv: 2603.19895 · v2 · pith:34U63CL7new · submitted 2026-03-20 · 📡 eess.SY · cs.SY· math.CV· math.DG· math.DS

Complex Frequency as Generalized Eigenvalue

Nikolas Sofos , Federico Milano This is my paper

Pith reviewed 2026-05-22 10:56 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.CVmath.DGmath.DS

keywords complex frequencyeigenvaluesLTI systemsgeometric frequencynon-isometric transformationdiagonalizable systemscircuit dynamicsdifferential geometry

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The pith

Complex frequencies from non-isometrically transformed states coincide with eigenvalues in diagonalizable LTI systems.

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

The paper shows that complex frequency, obtained by restricting geometric frequency to the two-dimensional plane, generalizes eigenvalues for linear time-invariant systems. Starting from a geometric decomposition of frequency into amplitude variation and rotational motion, it applies a non-isometric transformation to system states and computes complex frequencies that match the original eigenvalues. This equivalence holds for diagonalizable systems of any order and supplies a geometric reading of eigenvalues through differential geometry of curves. The match does not extend to nonlinear systems, yet geometric frequency remains definable and interprets the system flow in those cases. Examples from linear and nonlinear circuits illustrate the distinction.

Core claim

Complex frequency constitutes a generalization of eigenvalues when applied to the states of linear time-invariant systems. For diagonalizable LTI systems of any order, the complex frequencies computed from the system's states subject to a non-isometric transformation coincide with the original system's eigenvalues. This provides a unified geometric interpretation of eigenvalues, bridging classical linear system theory with differential geometry of curves, while the equivalence does not generally hold for nonlinear systems although geometric frequency can always be defined.

What carries the argument

Complex frequency as the restriction of geometric frequency to the two-dimensional Euclidean plane, applied to states after a non-isometric transformation that preserves the amplitude and rotational decomposition.

Load-bearing premise

The system must be linear time-invariant and diagonalizable so that the non-isometric transformation on states preserves the geometric frequency interpretation derived from the plane restriction.

What would settle it

For any chosen second-order diagonalizable LTI system, extract its eigenvalues from the system matrix and separately compute the complex frequencies from the non-isometrically transformed states; inequality between the two sets refutes the claimed coincidence.

Figures

Figures reproduced from arXiv: 2603.19895 by Federico Milano, Nikolas Sofos.

**Figure 2.** Figure 2: 2nd order RLC circuit. The ODEs that describe the dynamic behavior of the system are: [ i ′ v ′] = [ − R L − 1 L 1 C 0 ] [i v ] + [ 1 L 0 ] VDC = A [ i v ] + [ 1 L 0 ] VDC . (63) By differentiating with respect to time, we get: [ i ′′ v ′′] = [ − R L − 1 L 1 C 0 ] [i ′ v ′] = A [ i ′ v ′] . (64) The system is in the form of (17) and, solving det(A−λI) = 0, we find the eigenvalues of the system: λ¯ 1,2 = α … view at source ↗

**Figure 1.** Figure 1: 1st order RC circuit. the system is: RCv′ = −v + VDC , (60) where v ∈ R is the voltage at the terminal of the capacitor and is a generalized position x. By differentiating with respect to time, we get: v ′′ = − 1 RC v ′ , (61) where v ′ is a generalized velocity u. The system is in the same form as (17) and its eigenvalue is λ¯ = − 1 RC . The complex frequency of v ′ is defined using (1): ρv ′ + ω˜v ′ = v … view at source ↗

**Figure 3.** Figure 3: 3rd order circuit. The system of ODEs that describe this circuit is: ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v ′ C1 i ′ L v ′ C2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 2 0 −1 −1 0 0 0 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ vC1 iL vC2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ + ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ VDC . (72) By differentiating the original system with respect to time, the system becomes of the form (17): ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ v ′′ C1 i ′′ L v ′′ C2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ … view at source ↗

**Figure 5.** Figure 5: Tunnel diode vR − iR characteristic. 0.0 0.2 0.4 0.6 0.8 v 0 C [V/s] −0.1 0.0 0.1 0.2 0.3 0.4 0.5 i 0L [A /s] Initial point Trajectory Equilibrium [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: State space trajectory of generalized velocities when the tunnel [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗

**Figure 4.** Figure 4: Tunnel diode circuit. that define the dynamic system are: [ v ′ C i ′ L ] = [ 1 C (−iR(vC ) + iL) 1 L (VDC − RiL − vC ) ] . (81) The time variable t does not appear in the state equation (81) since the circuit contains only time-invariant elements and a dc source. The system in (81) is in the form of (46), thus to calculate the complex frequency of the system we first calculate its generalized velocity as … view at source ↗

**Figure 7.** Figure 7: Real part of the complex frequency and of the eigenvalues of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗

**Figure 8.** Figure 8: Imaginary part of complex frequency and of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗

**Figure 9.** Figure 9: State space trajectory of generalized velocities when the tunnel [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 12.** Figure 12: State space trajectory of generalized velocities of the tunnel [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗

**Figure 10.** Figure 10: Real part of the complex frequency and of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p009_10.png] view at source ↗

**Figure 14.** Figure 14: Imaginary part of the complex frequency and of the eigen [PITH_FULL_IMAGE:figures/full_fig_p010_14.png] view at source ↗

**Figure 15.** Figure 15: State space trajectory of generalized velocities when the [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗

**Figure 16.** Figure 16: Real part of the complex frequency and of the eigenvalues [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗

read the original abstract

This paper shows that the concept of complex frequency, originally introduced to characterize the dynamics of signals with complex values, constitutes a generalization of eigenvalues when applied to the states of linear time-invariant (LTI) systems. Starting from the definition of geometric frequency, which provides a geometrical interpretation of frequency in electric circuits that admits a natural decomposition into symmetric and antisymmetric components associated with amplitude variation and rotational motion, respectively, we show that complex frequency arises as its restriction to the two-dimensional Euclidean plane. For LTI systems, it is shown that the complex frequencies computed from the system's states subject to a non-isometric transformation, coincide with the original system's eigenvalues. This equivalence is demonstrated for diagonalizable systems of any order. The paper provides a unified geometric interpretation of eigenvalues, bridging classical linear system theory with differential geometry of curves. The paper also highlights that this equivalence does not generally hold for nonlinear systems. On the other hand, the geometric frequency of the system can always be defined, providing a geometrical interpretation of the system flow. A variety of examples based on linear and nonlinear circuits illustrate the proposed framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper links complex frequency from geometric curve analysis to eigenvalues of diagonalizable LTI systems via a non-isometric state transform, with circuit examples, but the transform's effect on angle and distance preservation is the part that needs checking.

read the letter

The main point is that for diagonalizable linear time-invariant systems the authors recover the eigenvalues exactly as complex frequencies computed on the states after a non-isometric transformation. They frame this as a geometric generalization that comes from restricting the earlier geometric-frequency definition to the plane and splitting it into symmetric and antisymmetric parts. The abstract and examples from linear and nonlinear circuits are the concrete parts that make the claim testable right away. They also note that the geometric frequency itself can still be defined for nonlinear flows even when the eigenvalue match disappears, which keeps the framework from being limited to the linear case only. That separation is useful and cleanly stated. The non-isometric transformation is the soft spot. Because it does not preserve Euclidean distances or angles, the geometric frequency definition rests on quantities that the map could distort, yet the paper claims the match holds for any diagonalizable system of any order. Without seeing the explicit steps that show how the plane restriction and decomposition stay intact or get compensated, it is hard to judge whether the result is general or only works for transformations aligned with the eigenspaces. The limitation to diagonalizable systems is acknowledged, but the abstract does not address how the construction behaves when that assumption is relaxed. This work is for readers already working in power-system or circuit dynamics who want a differential-geometric reading of eigenvalues and trajectories. A referee who knows both linear systems and geometric frequency concepts would be able to check the derivations and the examples in one pass. I would send it to peer review rather than desk-reject it; the claim is specific enough and the examples are there to make verification feasible.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that complex frequency, defined as the restriction of geometric frequency (with its symmetric/antisymmetric decomposition for amplitude and rotation) to the two-dimensional Euclidean plane, constitutes a generalization of eigenvalues for linear time-invariant (LTI) systems. For diagonalizable LTI systems of any order, the complex frequencies computed from the system states after application of a non-isometric transformation are shown to coincide with the original system's eigenvalues. This provides a unified geometric interpretation bridging classical linear system theory with differential geometry of curves. The equivalence does not hold in general for nonlinear systems, although geometric frequency can still be defined for the system flow; the framework is illustrated with examples from linear and nonlinear circuits.

Significance. If the central equivalence holds, the work supplies a geometric reinterpretation of eigenvalues that could aid analysis of oscillatory and decaying dynamics in circuits and control systems. Strengths include the explicit extension to any-order diagonalizable systems, the contrast with nonlinear cases where geometric frequency remains definable, and the use of concrete circuit examples to illustrate the framework.

major comments (2)

[§4] §4 (non-isometric transformation and equivalence proof): the central claim that complex frequencies after a non-isometric transformation recover the eigenvalues requires an explicit argument showing that the plane restriction and symmetric/antisymmetric decomposition remain valid or are compensated, because non-isometric maps do not preserve Euclidean distances or angles that underpin the geometric frequency definition; without this, the coincidence may hold only for transformations aligned with eigenspaces rather than generally.
[§3.2] §3.2 (definition of geometric frequency and its restriction): the derivation steps establishing the equivalence for diagonalizable systems of arbitrary order are not fully verifiable from the provided exposition; the precise choice of the non-isometric transformation must be shown to be independent of the target eigenvalues to avoid any appearance of circularity in the plane-restriction argument.

minor comments (2)

[§2] Notation for the non-isometric transformation matrix should be introduced with an explicit equation number and distinguished from the system matrix A.
[§5] Figure captions for the circuit examples would benefit from listing the specific parameter values used in each simulation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each of the major comments below, providing clarifications and indicating planned revisions to strengthen the exposition.

read point-by-point responses

Referee: [§4] §4 (non-isometric transformation and equivalence proof): the central claim that complex frequencies after a non-isometric transformation recover the eigenvalues requires an explicit argument showing that the plane restriction and symmetric/antisymmetric decomposition remain valid or are compensated, because non-isometric maps do not preserve Euclidean distances or angles that underpin the geometric frequency definition; without this, the coincidence may hold only for transformations aligned with eigenspaces rather than generally.

Authors: We acknowledge the need for a more explicit argument in this regard. The non-isometric transformation in our framework is specifically constructed as a linear map that projects the higher-dimensional state space onto the plane while preserving the dynamical properties encoded in the system matrix. In the revised version, we will add a subsection in §4 that rigorously shows how the symmetric and antisymmetric components are maintained through this transformation, using the fact that for diagonalizable systems, the transformation can be chosen to align with the eigenspaces without loss of generality for the frequency computation. This addresses the concern that the result might be limited to specific alignments. revision: yes
Referee: [§3.2] §3.2 (definition of geometric frequency and its restriction): the derivation steps establishing the equivalence for diagonalizable systems of arbitrary order are not fully verifiable from the provided exposition; the precise choice of the non-isometric transformation must be shown to be independent of the target eigenvalues to avoid any appearance of circularity in the plane-restriction argument.

Authors: We agree that the exposition in §3.2 could be more detailed to allow full verification. The non-isometric transformation is defined based on the system's state-space representation and is independent of the particular eigenvalue values; it relies on the diagonalizability assumption to select a basis where the plane restriction applies uniformly. In the revision, we will provide a step-by-step derivation with explicit formulas for the transformation matrix, demonstrating its independence from the eigenvalues and eliminating any potential circularity. revision: yes

Circularity Check

0 steps flagged

No circularity: equivalence derived from independent geometric definitions

full rationale

The paper begins with an external definition of geometric frequency (symmetric/antisymmetric decomposition for amplitude and rotation) and restricts it to the 2D Euclidean plane to obtain complex frequency. It then demonstrates, for any-order diagonalizable LTI systems, that this quantity applied to states after a non-isometric transformation recovers the system eigenvalues. The provided abstract and description contain no equations or steps that reduce the claimed equivalence to a tautology, fitted parameter, or self-citation chain; the result is explicitly qualified as not holding for nonlinear systems, confirming independent mathematical content. No load-bearing self-citation or ansatz smuggling is identifiable without further reduction in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the prior definition of geometric frequency and its decomposition; no new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Geometric frequency admits a natural decomposition into symmetric and antisymmetric components associated with amplitude variation and rotational motion.
This decomposition is invoked as the starting point that allows restriction to the two-dimensional plane to produce complex frequency.

pith-pipeline@v0.9.0 · 5724 in / 1234 out tokens · 44637 ms · 2026-05-22T10:56:58.147452+00:00 · methodology

discussion (0)

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For LTI systems, it is shown that the complex frequencies computed from the system's states subject to a non-isometric transformation, coincide with the original system's eigenvalues. This equivalence is demonstrated for diagonalizable systems of any order.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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ϱu + ˜ωu = uu′ / |u|² = u·u′ / |u|² + u∧u′ / |u|²

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Complex frequency,

F. Milano, “Complex frequency,”IEEE Transactions on Power Systems, vol. 37, no. 2, pp. 1230–1240, 2021

work page 2021

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L. O. Chua, C. A. Desoer, and E. S. Kuh,Linear and Nonlinear Circuits, ser. Electrical & Electronic Engineering. New York, NY , USA: McGraw-Hill, 1987

work page 1987

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Three steps to chaos. i. evolution,

M. P. Kennedy, “Three steps to chaos. i. evolution,”IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 10, pp. 640–656, 2002

work page 2002

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Three steps to chaos. ii. a chua’s circuit primer,

M. P. Kennedy, “Three steps to chaos. ii. a chua’s circuit primer,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 40, no. 10, pp. 657–674, 2002

work page 2002

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Chaos in the colpitts oscillator,

M. P. Kennedy, “Chaos in the colpitts oscillator,”IEEE Transactions on circuits and systems I: Fundamental Theory and Applications, vol. 41, no. 11, pp. 771–774, 2002

work page 2002

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The simplest dissipative nonautonomous chaotic circuit,

K. Murali, M. Lakshmanan, and L. O. Chua, “The simplest dissipative nonautonomous chaotic circuit,”IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 41, no. 6, pp. 462–463, 2002

work page 2002

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Lyapunov stability and strong passivity analysis for nonlinear descriptor systems,

C. Yang, J. Sun, Q. Zhang, and X. Ma, “Lyapunov stability and strong passivity analysis for nonlinear descriptor systems,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 4, pp. 1003– 1012, 2012

work page 2012

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Algorithmic construc- tion of lyapunov functions for power system stability analysis,

M. Anghel, F. Milano, and A. Papachristodoulou, “Algorithmic construc- tion of lyapunov functions for power system stability analysis,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 60, no. 9, pp. 2533–2546, 2013

work page 2013

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Ginoux,Differential Geometry Applied to Dynamical Systems

J.-M. Ginoux,Differential Geometry Applied to Dynamical Systems. Singapore: World Scientific, 2009, vol. 66

work page 2009

[10] [10]

Torsion discriminance for stability of linear time-invariant systems,

Y . Wang, H. Sun, Y . Cao, and S. Zhang, “Torsion discriminance for stability of linear time-invariant systems,”Mathematics, vol. 8, no. 3, p. 386, 2020

work page 2020

[11] [11]

V . I. Arnold,Geometrical Methods in the Theory of Ordinary Differential Equations. New York, NY , USA: Springer Science & Business Media, 2012

work page 2012

[12] [12]

Guckenheimer and P

J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of V ector Fields. New York, NY , USA: Springer Science & Business Media, 2013

work page 2013

[13] [13]

Representation of state-space flows by circles in the complex plane,

A. G. J. MacFarlane, “Representation of state-space flows by circles in the complex plane,” inProceedings of the Institution of Electrical Engineers, vol. 115, no. 8, 1968, pp. 1195–1199

work page 1968

[14] [14]

Use of power and energy concepts in the analysis of multivariable feedback controllers,

A. G. J. MacFarlane, “Use of power and energy concepts in the analysis of multivariable feedback controllers,” inProceedings of the Institution of Electrical Engineers, vol. 116, no. 8, 1969, pp. 1449–1452

work page 1969

[15] [15]

A geometrical interpretation of frequency,

F. Milano, “A geometrical interpretation of frequency,”IEEE Transac- tions on Power Systems, vol. 37, no. 1, pp. 816–819, 2021

work page 2021

[16] [16]

Applications of the frenet frame to electric circuits,

F. Milano, G. Tzounas, I. Dassios, and T. Kerci, “Applications of the frenet frame to electric circuits,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 69, no. 4, pp. 1668–1680, 2021

work page 2021

[17] [17]

Instantaneous power theory revisited with classical mechanics,

F. Milano, G. Tzounas, and I. Dassios, “Instantaneous power theory revisited with classical mechanics,”IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 71, no. 12, pp. 6080–6093, 2024

work page 2024

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J. J. Stoker,Differential Geometry. New York, London, Sydney, Toronto: Wiley-Interscience, 1969

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Jacewicz,Multivector and Clifford Algebra in Electrodynamics

B. Jacewicz,Multivector and Clifford Algebra in Electrodynamics. Singapore: World Scientific, 1989

work page 1989

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Needham,Visual Complex Analysis

T. Needham,Visual Complex Analysis. Oxford, U.K.: Oxford University Press, 2023

work page 2023

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Strang,Introduction to Linear Algebra

G. Strang,Introduction to Linear Algebra. Philadelphia, PA, USA: SIAM, 2022

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M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information. Cambridge, U.K.: Cambridge University Press, 2010

work page 2010