<p>Concatenating matrices is a common technique for uncovering shared structures in data through singularvalue decomposition (SVD) and low-rank approximations. A fundamental question arises: How does the singular-value spectrum of a concatenated matrix relate to the spectra of its individual components? In the present work, we develop a perturbation technique that extends classical results, such as Weyl’s inequality, to concatenated matrices. We establish analytic bounds that quantify the stability of singular values under small perturbations in submatrices. The results demonstrate that if submatrices are close in a norm, the dominant singular values of the concatenated matrix remain stable and enable controlled trade-offs between accuracy and compression. These provide a theoretical basis for improved matrix clustering and compression strategies with applications in the numerical linear algebra, signal processing, and data-driven modeling.</p>