Imagine discovering an ancient treasure chest sealed with a complex dual-lock mechanism, requiring two keys that must work together in a precise way. A matrix—a rectangular array of numbers—is like a locked chest holding valuable information that helps us understand the world around us. Matrices need keys like decompositions, which break them down into simpler components while preserving their essential properties, to help us understand them better. At times, special kinds of decompositions are required to have a deeper understanding of matrices.

Researchers in mathematics have uncovered a new approach to matrix decomposition, which could pave the way for significant advances in areas such as signal and image processing, machine learning, and speaker recognition.

Drs. Agnes Paras and Jenny Salinasan of the University of the Philippines – Diliman College of Science’s Institute of Mathematics (UPD-CS IM), along with Dr. Dennis Merino of Southeastern Louisiana University, studied the ϕS polar decomposition—a specialized form of polar decomposition.

“There are many ways to decompose a given matrix, but in the event that a prescribed decomposition is not always possible, the challenge is to obtain necessary and sufficient conditions for a prescribed decomposition to exist,” the authors explained.

Their study identified three conditions to determine whether a square matrix X has a ϕS​ polar decomposition: (1) the matrix product ϕS(X)X must have a square root that exhibits a specific symmetry; (2) ϕS(X)X and another matrix product, XϕS(X), must have the same fundamental properties; and (3) the matrices [XϕS(X)]kX must have even rank for any nonnegative integer k.

By identifying these conditions, the mathematicians discovered when a square matrix can be broken down into special types of matrices called symplectic and skew-Hamiltonian. “Symplectic matrices have applications in quantum optics, particularly, in the analysis of squeezed states of light, while skew-Hamiltonian matrices have applications in systems and control theory,” the authors added.

The authors noted that previous mathematical research such as that from de la Cruz and Teretenkov, had already provided conditions for complex matrices. “However, the conditions given in the complex case are not sufficient over an arbitrary field, while the conditions given in the real case are not necessary,” they added.

Their study, “The ϕS polar decomposition when S is skew-symmetric,” was published in Linear Algebra and its Applications, a journal that publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra. The research was supported by the UP Diliman Natural Sciences Research Institute.