In this paper, we consider the generalized low-rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $\underset{X}{min}\sum _{i=1}^m{\Vert A_i-B_{i}X{B}_i^T\Vert }_F^2,$ where $X$ is an unknown symmetric positive semidefinite matrix whose rank is less than or equal to a positive integer $k$. We first characterize the feasible set as $X=Y{Y}^T$, where $Y$ has the order $n\times k$, and then convert the generalized low-rank approximation into an unconstrained generalized optimization problem. Finally, we employ the nonlinear conjugate gradient method with an exact line search to solve the generalized optimization problem. We also give numerical examples to exemplify the results.

This paper establishes solvability conditions and general solution sets for a constrained Sylvester-type system over the commutative quaternion ring by proposing two distinct methods. As a key application, we also analyze the minimum solution of a related optimization problem when this system is solvable. Additionally, the solvability condition and the general form of the Hermitian solutions for this system over the commutative quaternion ring are established in this paper. The main results are validated through an algorithm and a numerical example.

This study primarily investigated the equivalence conditions for the existence of solutions to two quaternion matrix systems under constraints, as well as their general solutions. As an application, it focused on studying the reducibility of solutions to classical matrix equations and their applications in image processing, such as image encryption and decryption. Finally, an example is provided to validate the main results presented in this paper.