<p>Matrix computations have a long history of study in view of numerous applications in scientific and engineering research fields. Nowadays, it is important to develop numerical algorithms for observation matrices contaminated with random noise arising in research fields such as data science and machine learning. This study presents a unified view of considerable statistical applications relevant to the singular value decomposition with orthogonal projections for the observation matrices. In particular, we focus on the errors-in-variables linear regression model and the dynamic mode decomposition with observation errors to clarify important matrix structures of statistical estimators. From this perspective, we provide a general framework for consistency analysis of statistical estimators computed from the singular value decomposition using orthogonal projections. Our asymptotic matrix analysis leads to a basic principle for the construction of consistent estimators using the orthogonal projections, resulting in an extension of the existing estimators. Moreover, strong consistency of the estimators can be proved straightforwardly under reasonable conditions. It is worth noting that our framework of consistency analysis covers a complicated structured problem related to the Vandermonde matrices that arise in important real-world applications.</p>