blockbidiagonal
Blockbidiagonal refers to a matrix structure organized in blocks where nonzero blocks appear only on the main block diagonal and on one adjacent block diagonal, forming a two-band pattern. This generalizes the scalar bidiagonal form to matrices composed of small blocks. A block upper bidiagonal matrix, for example, has the form A = [B1 C1 0 … 0; 0 B2 C2 … 0; …; 0 0 … Bn], where Bi and Ci are p×p blocks and the zeros fill the remaining positions. A block lower bidiagonal variant replaces the superdiagonal blocks Ci with subdiagonal blocks Di. The total size of such a matrix is n·p by n·p, with n blocks along the main diagonal and p the size of each block.
Properties and computation: The block bidiagonal structure implies a sparsity pattern that enables efficient numerical algorithms.
Applications: Block bidiagonal matrices arise in discretizations of two-dimensional problems, such as finite-difference schemes for Poisson-type
Example: For n = 3 and block size p, an upper block bidiagonal matrix has nonzeros only in