Complex-Valued Linear Dynamical Systems

• CLDS is a dynamical system defined by complex state, input, and output variables with coefficients operating via bimatrix algebra.
• It is applied in control, quantum mechanics, signal processing, and neuroscience to analyze stability, controllability, and feedback design.
• CLDS supports efficient learning through spectral filtering and closed-form solutions, advancing high-dimensional and oscillatory system research.

A Complex-Valued Linear Dynamical System (CLDS) is a dynamical system whose state, input, and output variables evolve in complex vector spaces under linear transformations. Such systems generalize classical real-valued linear dynamical systems by allowing both the coefficients and variables to be complex, naturally modeling a wide range of phenomena in control theory, quantum systems, signal processing, and network neuroscience. The canonical forms of CLDS cover models whose evolution may depend either on the state alone or on both the state and its complex conjugate, with the distinction formalized using bimatrix algebra. CLDS theory unifies the treatment of normal (real/complex) linear systems, antilinear systems, and models arising from the complexification of physical, biological, and engineered systems.

1. Canonical Forms and Mathematical Structure

The general discrete-time CLDS is written as

$x(k+1) = \{A_1, A_2\}\, x(k) + \{B_1, B_2\}\, u(k)$

where $x\in\mathbb C^n$, $u\in\mathbb C^m$, and $\{A_1, A_2\}\, x = A_1 x + A_2 \bar{x}$ is the bimatrix action; similarly for inputs. This covers both:

• The normal linear system: $A_2 = 0$, $B_2 = 0$
• The antilinear system: $A_1 = 0$, $B_1 = 0$

In continuous time, and in input-output notation,

$$\dot{x}(t) = \{A_1, A_2\} x(t) + \{B_1, B_2\} u(t), \quad y(t) = \{C_1, C_2\}x(t) + \{D_1, D_2\}u(t)$$

As a special case, the model may ignore conjugate terms:

$$x_{t+1} = A x_t + B u_t, \quad y_t = C x_t + D u_t, \quad A,B,C,D \in \mathbb C$$

For quantum dynamical systems or fields, these models align closely with linearized quantum evolution and structured state space models (Hazan et al., 29 Jan 2026).

Two mathematically equivalent real representations are frequently used:

• Real-embedding: Map $$x \mapsto [\text{Re} x; \text{Im} x] \in \mathbb R^{2n}$$, translating system evolution into a $2n$-dimensional real linear system.
• Complex lifting: Map $x \mapsto [x; \bar{x}] \in \mathbb C^{2n}$, yielding a "doubled" system especially suited for analysis via bimatrix operations (Zhou, 2017, Zhou, 2017).

Bimatrix algebra underpins much of the theory: addition, multiplication, inversion, and conjugate transpose of bimatrices are well-defined, leading to compact analytic statements of controllability, observability, stability, and feedback design (Zhou, 2017).

2. Controllability, Observability, and Stability

Generalizations of the PBH test describe controllability for CLDS:

$$\forall s\in \mathbb C: \, \operatorname{rank}\!\begin{bmatrix} sI_n - A_1 & -A_2 \ -B_2^\# & sI_n - A_1^\# \ B_1 & B_2 \ B_2^\# & B_1^\# \end{bmatrix} = 2n$$

with observability tests similarly formulated. The eigenvalues of a CLDS are those of its real or complex-lifted representation, ensuring the closed-loop spectrum is symmetric with respect to the real axis.

Stability is defined via Hurwitz (continuous time) or Schur (discrete time) criteria for the real-representation matrix. Lyapunov stability theory extends with the Lyapunov bimatrix equations, whose solution ensures asymptotic stability:

• Continuous-time: $$\{A_1, A_2\}^H \{P_1, P_2\} + \{P_1, P_2\} \{A_1, A_2\} = -\{Q_1, Q_2\}$$.
• Discrete-time: $$\{A_1, A_2\}^H \{P_1, P_2\} \{A_1, A_2\} - \{P_1, P_2\} = -\{Q_1, Q_2\}$$. Closed-form solutions employ block diagonalizations or Neumann series expansions (Zhou, 2017).

3. System Identification, Control Design, and Pole Assignment

The pole assignment and optimal control of CLDS are structured via solutions to bimatrix equations:

• Generalized Sylvester Bimatrix Equations for pole placement: Full feedback $u = \{K_1, K_2\} x$ can assign any spectrum symmetric about the real axis, provided a nonsingular solution exists (Zhou, 2017).
• State Feedback LQR: The infinite-horizon LQR problem is characterized by a bimatrix algebraic Riccati equation whose solution yields the unique optimal gain:

$$\{P_1, P_2\} = \{Q, 0\} + \{A_1, A_2\}^H \{P_1, P_2\} \{A_1, A_2\} - \{A_1, A_2\}^H \{P_1, P_2\} \{B_1, B_2\} \{S_1, S_2\}^{-1} \{B_1, B_2\}^H \{P_1, P_2\} \{A_1, A_2\}$$

where $$\{S_1, S_2\} = \{R, 0\} + \{B_1, B_2\} \{P_1, P_2\} \{B_1, B_2\}$$ (Zhou, 2017). Monotonic iterative algorithms guarantee convergence to the unique positive definite solution.

Two closed-loop design schemes for the antilinear case (where $A_1=B_1=0$) are available: anti-preserving (achieves an antilinear closed loop) and normalization (converts the closed-loop system to a normal linear form) (Zhou, 2017).

In application to time-delay and second-order physical systems, CLDS formulations support controller synthesis where standard real-valued approaches would become cumbersome (Zhou, 2017, Zhou, 2017).

4. CLDS in Network Neuroscience and Analytic Signal Lifting

Recent work has embedded observed neural signals into a complex field by augmenting real BOLD data $x(t)\in\mathbb R^N$ with conjugate momentum estimates $p(t)\in\mathbb R^N$, forming $\psi(t)=x(t)+i\,p(t)$. For quadratic Hamiltonians, the evolution follows Hamilton's equations:

$\frac{dq}{dt} = -H p, \qquad \frac{dp}{dt} = H q$

with $H$ symmetric and positive-definite, leading to a Schrödinger-like linear CLDS:

$i\,\frac{d\psi}{dt} = H\psi \,.$

Empirically, optimal $p(t)$ is given by the Hilbert transform of $x(t)$. The resulting analytic state captures dynamic modes, intrinsic timescales, and structure-function coupling in brain networks far more accurately than real-valued models (short-horizon prediction $r: 0.12\to0.82$) (Zhang et al., 29 Sep 2025).

When extended to nonlinear, input-driven, and non-equilibrium regimes, small skew-symmetric and dissipative components in $H$ model nonconservative, biological irreversibility. The approach elucidates task-evoked coupling reconfiguration, information propagation backbones, and age-related changes in network connectivity.

5. CLDS in Quantum Systems and High-Dimensional Learning

CLDS foundationally describes the linear-response evolution of quantum systems. Here, sector-bounded spectra naturally arise: if $A$ is the system matrix, $$\operatorname{Spec}(A) \subset \mathbb{C}_\beta = \{z: |z|\leq1, |\arg(z)|\leq \beta\}$$, encapsulating stability and frequency constraints.

Efficient learning of quantum CLDS is enabled by Quantum Spectral Filtering, which compresses state trajectories using Discrete Prolate Spheroidal Sequences (DPSS/Slepian basis) to form a filter bank spanning the effective quantum dimension $k^* = \lceil \frac{\beta}{\pi} W \rceil$ for memory horizon $W$ and spectral band $\beta$. Learning and sample complexity become independent of the ambient state dimension—crucial for high-dimensional quantum or oscillatory systems (Hazan et al., 29 Jan 2026).

Algorithmically:

• The quantum information matrix $Z_W(\beta)$ is constructed and diagonalized.
• Trajectories are projected onto the span of the top $k^*$ DPSS vectors.
• Online learners operate in this compressed subspace, achieving oracle rates for prediction error.

Applications extend to structured state-space models and are relevant for oscillatory non-quantum time series (e.g., EEG, speech), with the effective dimension $k^*$ determinative of achievable accuracy.

6. Computation and Transient Dynamics in Complex-Valued Neural Systems

CLDSs provide a framework for recurrent complex-valued neural networks with explicit, closed-form solutions:

$\dot{x}(t) = (i\omega I + K) x(t)$

where $K = \epsilon e^{-i\phi} A$ encodes coupling and time delays. These systems produce rich transient spatiotemporal patterns, including chimera states—partially synchronized configurations observed in both biological and synthetic networks (Budzinski et al., 2023).

Key computational primitives include:

• Dynamics-driven logic gates via targeted input weightings and readout mappings.
• Short-term memory realized by steering the state along prescribed phase patterns.
• Secure message passing through network-encrypted input encodings.
• Direct biological decoding: living neurons can decode CLDS spatiotemporal signatures with high accuracy when injected with appropriate current patterns.

The exact analytical tractability and capacity for implementing combinatorial computations in linear CLDSs support their utility in hybrid computing and signal-processing architectures.

7. Broader Implications and Research Directions

CLDS theory consolidates multiple subfields:

• Quantum control: Formulating dissipative and coherent control.
• Delay and second-order systems: Allowing dense real-to-complex transformations for efficient control design.
• Neuroscience and signal analysis: Complex lifting reveals intrinsic dynamic hierarchies, structure-function alignment, and biologically plausible connectivity.
• Efficient learning: Dimension-free identification and prediction via spectral filtering.

The bimatrix algebra and real-representation techniques developed in foundational works (Zhou, 2017, Zhou, 2017) remain central in ongoing developments. Current research explores nonlinear extensions, non-Hermitian couplings, sparse network recovery in high dimensions, and biologically realistic learning algorithms.

A plausible implication is that, due to their closed-form tractability, ability to encode memory and logic, and structural interpretability, CLDSs are foundational modeling tools in both the theoretical analysis of complex systems and the construction of efficient, interpretable machine learning and signal processing pipelines.