Strong Eigenvalues in Spectral Theory

• Strong eigenvalues are fundamental spectral characteristics that define extreme stability conditions and strict positivity across elasticity, random matrices, and graph theory.
• They are used to verify material stability in elasticity, ensure almost-sure convergence in large random matrices, and construct sharp eigenvalue bounds in structured graphs.
• They also strengthen numerical analysis by enabling robust computations and enhance asymptotic spectral theory through strong coupling and spectral edge investigations.

Strong eigenvalues arise as a central concept across several domains in mathematics and mathematical physics, particularly in the spectral theory of matrices, elasticity, graph theory, and random matrix theory, often reflecting extremal spectral phenomena, stability under perturbations, or strict positivity conditions. The term “strong eigenvalue” acquires precise technical meanings in context, linked to strong ellipticity in elasticity tensors, strong convergence of extreme eigenvalues in probability, spectral stability in numerical analysis, or the identification of spectral edges and outlier phenomena (“strong spikes”) in high-dimensional random and structured systems.

1. Strong Eigenvalues and Strong Ellipticity in Tensors

In elasticity theory, the M-eigenvalue problem provides a rigorous formulation for “strong” eigenvalues via the fourth-order elasticity tensor $C_{ijkl}$. The M-eigenvalues $\theta$ are defined as solutions to

$$C_{ijkl} y_j x_k y_l = \theta x_i,\qquad x_i x_i = y_j y_j = 1,$$

with analogous dual equations exchanging $x$ and $y$ (Xiang et al., 2017). These eigenvalues are invariant under orthogonal transformations and hence are intrinsic characteristics of the material’s elastic properties.

Crucially, a stiffness tensor is strongly elliptic if and only if all M-eigenvalues are positive: strong ellipticity $\iff \min\theta > 0$. This equivalence provides necessary and sufficient conditions for material stability, directly linking spectral data (the strong eigenvalues) to physical requirements in the isotropic, cubic, polar anisotropic, tetragonal, and orthotropic symmetry classes. For instance, in the isotropic case, the M-eigenvalues reduce to the shear modulus $\mu$ and the P-wave modulus $\lambda + 2\mu$, with positivity conditions $\mu > 0$, $\lambda + 2\mu > 0$ enforcing strong ellipticity.

2. Strong Convergence of Extreme Eigenvalues

In random matrix theory, “strong eigenvalues” often refers to the almost sure (a.s.) or strong limit of the extreme eigenvalues of large random matrices. For example, for a quaternion self-dual Hermitian Wigner matrix $H_n$, the normalized matrix $Q_n = H_n/\sqrt n$ satisfies

$$P\left( \lim_{n\to\infty} \lambda_{\max}(Q_n) = 2\sigma \right) = 1,$$

where $\sigma^2$ is the variance of the off-diagonal entries, provided the entries have zero mean, bounded variance, and bounded fourth moment (Yin et al., 2013). This “strong convergence” characterizes the almost-sure location of spectral edges, generalizing the classical Bai–Yin and Wigner results to quaternionic and further to non-Gaussian settings.

Analogous results hold for various covariance-related ensembles. For the symmetrized auto-cross covariance matrix $M_n$, the largest and smallest eigenvalues converge almost surely to the edges $\pm d(c)$ of the limiting spectral distribution (LSD), with $d(c)$ given explicitly in terms of the asymptotic dimension ratio $n/T\to c$ (Wang et al., 2013).

3. Extremal Spectral Bounds in Structured Graphs

“Strong” eigenvalue bounds are prominent in spectral graph theory, both in the sense of sharp extremal upper bounds and through explicit construction of graphs realizing (or closely attaining) these bounds. For a graph $G$ on $n$ vertices, the $k$th largest singular value $\lambda^*_k(G)$ satisfies

$\lambda^*_k(G) \le \frac{n}{2\sqrt{k-1}},$

and this upper bound is strict for all $k\ge3$ except for certain highly structured graphs (Nikiforov, 2015). The extremal constructions rely on Taylor’s strongly regular graphs and on Kharaghani’s symmetric $(-1,1)$-matrix constructions, with the associated classes $\mathbb S_{s^2}$ realizing equality in the upper bound for infinitely many $n$.

The table below summarizes key strong eigenvalue bounds for graphs:

Quantity	Bound	Achievability
$A_k(n)=\max_{v(G)=n}\lambda_k(G)$	$\dfrac{n}{2\sqrt{k-1}}$	Strict for $k\ge3$, approached by SRGs
$A^*_k(n)=\max_{v(G)=n}\lambda^*_k(G)$	$\dfrac{n}{2\sqrt{k-1}}$	Achieved for $k=s^2+1$ via $\mathbb S_{s^2}$

Strong bounds govern associated combinatorial parameters, such as clique and chromatic numbers, and enter the study of Nordhaus–Gaddum type problems and Ky Fan norms, further connecting spectral extremality to broader graph invariants.

4. Strong Spikes and Outlier Eigenvalues in Random Ensembles

In high-dimensional random matrices, “strong spikes” refer specifically to atypically large outlier eigenvalues that detach from the bulk of the spectral distribution due to high-rank perturbations (spiked models). For the log-concave ensemble $Q = T XX^* T^*$ (where $T$ encodes the spikes), under strong spike conditions $d_i > y+\varepsilon$ (with $y=M/N$), the outlier eigenvalues satisfy

$$\lambda_i(Q) = \theta(d_i) + \frac{d_i}{\sqrt N} \xi_i + O_\prec(N^{-1/2+\varepsilon}),$$

with deformed location $\theta(d_i) = 1 + d_i + y + y/d_i$ and asymptotically Gaussian fluctuations of order $N^{-1/2}$ (Bao et al., 2022). This regime contrasts with the Tracy–Widom universality of the spectral edge in the absence of strong spikes and is distinguished by the sensitivity of outliers to finite-rank deformations.

5. Spectral Edges and Delocalization in Sparse and Band Random Matrices

For sparse or band random matrices $X$ with bandwidth $W$ and independent, centered, variance-one entries with sub-exponential tails, the largest eigenvalue of $X/\sqrt{W}$ converges in probability to 2 as $N\to\infty$ provided

$W \gg (\log N)^{6(1+\alpha)},$

with $\alpha$ parameterizing the tails (Benaych-Georges et al., 2013). This strong spectral edge result generalizes Wigner’s law to non-homogeneous sparsity and provides a matching delocalization result: eigenvectors corresponding to edge eigenvalues cannot be localized on fewer than $W/\log N$ coordinates, a significant threshold for understanding strong eigenvalue-induced delocalization phenomena in random systems.

6. Strong Eigenvalues in Numerical Analysis and Stability

In computational linear algebra, “strong” eigenvalues are reflected in forward and backward numerical stability of eigenvalue computations, particularly for closed-form solutions in low dimensions. For real, diagonalizable $3\times3$ matrices $A$, numerically stable eigenvalue evaluation is achieved using four invariants: trace $I_1$, deviatoric invariants $J_2$, $J_3$, and the discriminant $\Delta$ (Habera et al., 31 Oct 2025). The explicit closed-form solution

$$\lambda_k = \frac{I_1}{3} + \frac{2}{3} \sqrt{3J_2}\cos\left( \frac{\varphi+2\pi(k-1)}{3} \right),\quad \varphi = \arctan\left( \frac{\sqrt{27\Delta}}{27 J_3} \right)$$

is forward-stable (i.e., errors do not grow significantly compared to input perturbations) for matrices with moderately conditioned eigenbases, sharply outperforming iterative QR-based methods in speed while maintaining comparable accuracy. The requirement of strong (well-conditioned) invariants directly ties the notion of “strong eigenvalue” to robust numerical performance.

7. Asymptotic Spectral Theory: Robin Laplacians and Strong Coupling

In spectral problems for differential operators, the term “strong” appears in the context of strong coupling asymptotics. For the Robin Laplacian in polygonal domains with large parameter $\alpha$, the leading eigenvalues separate into “corner-induced” and “side-induced” spectra. The first $K$ eigenvalues scale as $E_n(R^\Omega_\alpha) = E_n \alpha^2 + o(\alpha^2)$ due to domain geometry, while for $n > K$,

$$E_{K+n}(R^\Omega_\alpha) = -\alpha^2 + E_n(L_{\text{eff}}) + o(\alpha),$$

where $L_{\text{eff}}$ is an effective Schrödinger operator on the boundary (Khalile et al., 2018). The notion of “strong eigenvalues” thus encompasses both edge-induced spectral scaling and the emergence of geometric and boundary effects in the strong-coupling regime.

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In summary, “strong eigenvalues” formalize critical spectral phenomena encountered at the intersection of stochastic, algebraic, analytic, and physical frameworks. The manifestations range from necessary and sufficient spectral positivity in elasticity, almost-sure edge convergence in random matrices, sharp extremal bounds in graphs, outlier detection in spiked models, stable computational algorithms, and asymptotic splitting in PDE eigenvalue problems. In each context, strong eigenvalues demarcate transitions between qualitative spectral behaviors, ensure stability (either physical or numerical), or optimize extremal properties central to the structural analysis of complex systems.