We introduced an efficient technique to suppress speckle noise in medical ultrasound images while maintaining low computational complexity. The medical ultrasound image is divided into overlapping subimages, and Lanczos decomposition is then applied to the average Hermitian covariance matrix of all subimages. The resultant orthonormal vectors are used for filtering speckle noise through orthogonal and oblique projections, i.e., by projecting noisy signal onto the signal subspace. After sorting the orthonormal vectors, an orthogonal projection matrix is formed by selecting the first K vectors contributing to the signal, whereas an oblique projection matrix is formed by selecting the first K vectors contributing to the signal and the last K vectors contributing to the noise. The procedure of Lanczos is also followed with singular value decomposition (SVD). Schemes are applied to real ultrasound images and two types of speckle noise simulations: fine and rough speckle noise. Numerical and visual results depict that proposed technique has outperformed various popular benchmark schemes, i.e., Frost, Lee, probabilistic nonlocal means, geometric nonlinear diffusion filter, guided speckle reducing bilateral filter, and SVD, while maintaining a competitive computational complexity. Lanczos-based scheme showed a relatively lagging performance in terms of resolution, which was always outshone by a leading performance in terms of key measures such as signal-to-noise ratio (SNR), peak signal-to-noise ratio (PSNR), feature similarity, and mean structural similarity. Orthogonal and oblique projections were found to perform the same except when speckle noise is fine, where insignificant differences were found in terms of SNR, PSNR, and resolution. Lanczos-based scheme tends to offer a better estimation of orthonormal vectors than SVD, consequently a better speckle noise suppression. Furthermore, it offers an efficient tuning parameter per block/subimage size to treat various speckle noise patterns.