This paper proposes Singular Value Decomposition-based Least Squares (SVD-LS) as an efficient multi-class pneumonia classification framework for accurate and early diagnosis of pneumonia. It leverages powerful feature representations extracted from state-of-the-art self-supervised learning and transfer learning models and, instead of computationally expensive gradient-based fine-tuning, employs a closed-loop, non-iterative classification approach to ensure efficiency while maintaining accuracy. Experimental results demonstrate that SVD-LS achieves competitive performance while significantly reducing computational costs, making it a suitable alternative for real-time medical imaging applications. The source code is available at github.com/meterdogan07/SVD-LS.