Gradient Flow for Generalized Normal Equation with Applications to Linear Matrix Equations

Solving a system of linear equations by its normal equation usually is highly unrecommended because this approach worsens the condition number and inflates the computational cost. For linear systems whose unknowns are matrices, such as the Sylvester equation, Lyapunov equation, Stein equation, and a variate of their generalizations, the formulation of the corresponding normal equation in the sense of tensor operators offers a common structure via gradient dynamics. This paper explains the setting of this framework and demonstrates its versatility by one simple ODE integrator that can handle almost all these types of problems.