Rotational Polarity of Eigenvectors (RPE)

• Rotational Polarity of Eigenvectors (RPE) is a geometric and algebraic framework that distinguishes pure rotations from reflections in eigenvector matrices.
• It employs an orientation-correction algorithm with arctan2-based rotations to stabilize eigenbases in high-dimensional optimization and directional statistics.
• RPE is practically applied in random matrix theory, directional statistics, and neural network optimization by transforming embedded reflections into full 2π rotational domains.

Rotational Polarity of Eigenvectors (RPE) identifies and characterizes the orientation behavior of eigenvectors under linear transformations, specifically emphasizing the distinction between pure rotations and reflections in orthonormal eigenvector matrices and the dynamical evolution of curvature axes in optimization scenarios. RPE provides a geometric and algebraic framework for interpreting complex eigenvectors and their rotation senses, stabilizing basis directions in evolving eigendecompositions, and elucidating optimization-induced exploration mechanisms in loss landscapes. In practice, RPE unifies the detection of embedded reflections, transformation into extended rotational domains, and the statistical stabilization of evolving eigenbases, with notable applications in directional statistics, random matrix theory, and neural network optimization.

1. Mathematical Definition and Geometric Interpretation

Rotational Polarity of Eigenvectors (RPE) is defined for orthonormal matrices $V \in \mathbb{R}^{n \times n}$, where $V^T V = I$ and $\det V = \pm 1$. The sign of $\det V$ encodes the matrix’s handedness: $\det V = +1$ implies $V \in SO(n)$ (a proper rotation matrix), while $\det V = -1$ identifies $V \in O(n)\setminus SO(n)$, marking the presence of an odd number of reflections embedded within the transformation (Damask, 13 Feb 2024).

RPE is the process of detecting these “handedness flips” (embedded reflections) and converting them into equivalent pure rotations, ensuring the matrix product defines a basis strictly in $SO(n)$. This conversion extends angular representations from the previously restricted $[-\pi, \pi]$ domain to the full $2\pi$ domain, removing statistical ambiguities and wrap-around artifacts in directional data.

Geometrically, each eigenvector of a real anti-symmetric matrix with imaginary eigenvalues can be interpreted in real geometric algebra as a multi-component spinor—specifically, as a duplet (in 2D) or triplet (in 3D) of real vectors that undergo rigid-body rotation under the generator and its finite rotation operator (Hitzer, 2013). The sign of the eigenvalue (“polarity”) determines the direction of this rotation, directly encoding the sense of handedness—right-handed or left-handed rotation.

2. Orientation-Correction Algorithm for Basis Stabilization

To consistently orient an eigenvector basis in $SO(n)$, RPE employs a sequential subspace algorithm. At each step $k$, the $k$th column of the current matrix $V_{k-1}$ is optionally reflected (choosing sign $s_k = -1$ as needed) and rotated toward the canonical basis vector $\pi_k$ via a cascade of major-arc Givens rotations. The diagonal sign-matrix $S = \operatorname{diag}(s_1,\ldots,s_n)$ is chosen so that $\det V \cdot \det S = 1$, enforcing proper rotation (Damask, 13 Feb 2024).

The modified “arctan2-method” computes rotation angles for each subspace, guaranteeing the first rotation in each subspace spans the full $2\pi$ domain and subsequent subrotations occupy $(0,\pi)$. This algorithm provides closed-form orientation recovery in arbitrary dimension, eliminating hemisphere lock and improving the interpretability of pointing statistics. The key pseudocode steps sort eigenpairs, compute reflection signs, solve rotation angles $\theta_{k,j}$ by arctan2, construct cumulative rotation matrices, and transform the eigenbasis accordingly.

3. RPE in Geometric Algebra: Complex Eigenvectors and Polarity

For real anti-symmetric matrices $A$ in $\mathbb{R}^n$, nonzero eigenvalues manifest as imaginary-conjugate pairs $\lambda = \pm i \omega$. The corresponding complex eigenvectors are recast as geometric-algebra spinors, with each spinor’s vector components rotating rigidly in the plane defined by the generator bivector $I$ (Hitzer, 2013). In two dimensions, the action $A x = \lambda x$ with $A = a I_2$ and $I_2 = e_1 e_2$ yields duplet spinors $x_+, x_-$ whose vector pairs rotate by $+90^\circ$ and $-90^\circ$, fixing the rotational polarity.

Finite rotations via the Cayley transform, $$R(\theta) = \exp(A \theta/a) = \cos\theta + \sin\theta\, I$$, leave each spinor invariant up to a left phase $e^{\pm I \theta}$. In three dimensions, eigen-spinors project onto the plane orthogonal to the rotation axis and undergo analogous rigid-duplet rotation by $\pm\theta$ about the axis, with the sign unambiguously encoding handedness. Rotational polarity is thus the sign in the exponent governing the direction of spinor rotation under both infinitesimal and finite operators.

4. Dynamical RPE: Eigenvector Rotation in Optimization

In gradient descent (GD) and related optimization procedures, RPE quantifies how leading eigenvectors of the loss Hessian $H(\theta)$ rotate as training traverses the classical stability threshold $(\eta \lambda_1 = 2)$ (Wang et al., 16 Nov 2025). Below threshold, the top curvature axis aligns toward sharper directions; above threshold, rotations drive the axis away, promoting exploration of flatter landscape regions.

For parameterized models, relative changes in principal eigenvector ratios under GD reveal that instability induces rotational polarity reversals, explicitly partitioning learning rates into regimes of curvature concentration and flattening. The angle between successive top eigenvectors, $$RPE_t = \arccos(v_1(\theta_{t+1}) \cdot v_1(\theta_t))$$, directly tracks these rotations, with monotonic growth beyond critical instability and immediate reversal on learning rate reduction.

Repeated RPE steps systematically contract higher-order curvature moments toward lower values, establishing a provable flatness bias in GD. Theoretical results formalize contraction envelopes, median drift, and mass concentration, showing robust flattening as a function of instability duration and magnitude.

5. Temporal Stabilization and Directional Statistics

For applications involving temporally evolving eigenbases—such as adaptive filtering or time-series SVD—RPE offers two stabilization schemes (Damask, 13 Feb 2024):

• Dynamic stabilization: Applies causal filters to the sequence of eigenbases, averaging directions and renormalizing columns, followed by re-orientation using the arctan2-based method. This reduces angular variance (“wobble”) and produces basis coefficients with persistent directional statistics.
• Static stabilization: Shrinks uninformative noise modes by setting corresponding rotation angles to zero, reconstructing a modal-only basis via Givens cascades for informative subspaces. This approach improves out-of-sample correlation estimates and statistical persistence.

Empirical results on financial timeseries demonstrate that the modified orientation-correction algorithm cleanly identifies genuine directional excursions and separates signal from noise by anchoring modes associated with persistent eigenvalue concentration outside the Marčenko–Pastur bulk.

6. RPE in Deep Learning Optimizers and Open Research Directions

Adaptive optimization schemes (e.g., Adam, RMSprop) affect RPE dynamics by suppressing curvature in principal Hessian directions, often preventing instability-induced rotations and the associated flatness bias (Wang et al., 16 Nov 2025). Reintroducing controlled instabilities (“Clipped-Ada”) by capping preconditioner estimates restores RPE-driven rotations, yielding improved generalization.

Ongoing research explores extending RPE theory to non-diagonal and highly nonlinear network architectures, validating empirical metrics for curvature moments, and integrating RPE insights with other spectral regularization techniques. A plausible implication is that optimizers designed to directly amplify beneficial rotational exploration may offer convergence guarantees and enhanced generalization.

7. Summary Table: RPE Manifestations Across Domains

Domain/Context	RPE Manifestation	Reference
Eigenbasis Orientation	Reflection detection; major-arc rotation	(Damask, 13 Feb 2024)
Geometric Algebra	Spinor duplet/triplet rotation polarity	(Hitzer, 2013)
Optimization Dynamics	Flatness bias via eigenvector rotation	(Wang et al., 16 Nov 2025)
Temporal Statistics	Dynamic/static stabilization schemes	(Damask, 13 Feb 2024)

RPE reveals a unifying geometric principle: the sign structure and angular domain of eigenvector transformations systematically encode the handedness, rotation sense, and statistical persistence of evolving bases and optimization trajectories. This framework facilitates improved analysis, visualization, and algorithmic control in fields ranging from random matrix theory to deep learning.