Block Matrix - Block Tridiagonal Matrices

Block Tridiagonal Matrices

A block tridiagonal matrix is another special block matrix, which is just like the block diagonal matrix a square matrix, having square matrices (blocks) in the lower diagonal, main diagonal and upper diagonal, with all other blocks being zero matrices. It is essentially a tridiagonal matrix but has submatrices in places of scalars. A block tridiagonal matrix A has the form

\mathbf{A} = \begin{bmatrix} \mathbf{B}_{1} & \mathbf{C}_{1} & & & \cdots & & 0 \\ \mathbf{A}_{2} & \mathbf{B}_{2} & \mathbf{C}_{2} & & & & \\ & \ddots & \ddots & \ddots & & & \vdots \\ & & \mathbf{A}_{k} & \mathbf{B}_{k} & \mathbf{C}_{k} & & \\ \vdots & & & \ddots & \ddots & \ddots & \\ & & & & \mathbf{A}_{n-1} & \mathbf{B}_{n-1} & \mathbf{C}_{n-1} \\ 0 & & \cdots & & & \mathbf{A}_{n} & \mathbf{B}_{n} \end{bmatrix}

where Ak, Bk and Ck are square sub-matrices of the lower, main and upper diagonal respectively.

Block tridiagonal matrices are often encountered in numerical solutions of engineering problems (e.g., computational fluid dynamics). Optimized numerical methods for LU factorization are available and hence efficient solution algorithms for equation systems with a block tridiagonal matrix as coefficient matrix. The Thomas algorithm, used for efficient solution of equation systems involving a tridiagonal matrix can also be applied using matrix operations to block tridiagonal matrices (see also Block LU decomposition).

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