  The Kalman-Bucy filter is extensively utilized across various applications.
However, its computational complexity increases significantly in large-scale
systems. To mitigate this challenge, a low-rank approximated Kalman--Bucy
filter was proposed, comprising Oja's principal component flow and a
low-dimensional Riccati differential equation. Previously, the estimation error
was confirmed solely for linear time-invariant systems with a symmetric system
matrix. This study extends the application by eliminating the constraint on the
symmetricity of the system matrix and describes the equilibrium points of the
Oja flow along with their stability for general matrices. In addition, the
domain of attraction for a set of stable equilibrium points is estimated. Based
on these findings, we demonstrate that the low-rank approximated Kalman--Bucy
filter with a suitable rank maintains a bounded estimation error covariance
matrix if the system is controllable and observable.
