SCHUR DECOMPOSITION AND DCT-DST-BASED MATRIX-VECTOR MULTIPLICATION ALGORITHM FOR REAL BLOCK SKEW CIRCULANT MATRICES

Real block skew circulant matrices appear in structured linear systems and signal processing, where their algebraic properties can be exploited to design efficient numerical algorithms. This paper develops a Schur decomposition specifically for real block skew circulant matrices and proves that they can be block diagonalized via an orthogonal similarity transformation constructed from discrete cosine and discrete sine transform matrices. The resulting Schur form consists of low dimensional independent blocks, enabling a fast matrix-vector multiplication scheme based on DCT-DST operations rather than direct computation. A complexity analysis shows that the proposed approach reduces arithmetic cost while preserving numerical stability due to the orthogonality of the transforms. Numerical results validate the theoretical findings and demonstrate the efficiency of the method for large scale problems.