blokkdiagonális

A "blokkdiagonális" is a mathematical matrix that has non-zero elements only in square submatrices along the main diagonal, with all other elements being zero. These non-zero submatrices are called blocks, and the structure is analogous to a diagonal matrix, where the diagonal elements are themselves matrices.

The general form of a blokkdiagonális matrix A of size m x m can be represented as:

A = diag(A1, A2, ..., Ak)

where A1, A2, ..., Ak are square matrices, and diag(...) denotes the arrangement of these matrices along the main diagonal. The total size m is the sum of the sizes of these individual blocks. For example, if A1 is of size n1 x n1, A2 is of size n2 x n2, ..., and Ak is of size nk x nk, then m = n1 + n2 + ... + nk.

Blokkdiagonális matrices have several useful properties. Operations such as addition, multiplication, and inversion can be performed block-wise, simplifying calculations. For instance, the determinant of a blokkdiagonális matrix is the product of the determinants of its diagonal blocks. Similarly, eigenvalues are the union of the eigenvalues of the individual blocks.

These matrices are commonly encountered in various fields, including linear algebra, numerical analysis, and computer science. They arise in the solution of systems of differential equations, in the study of Markov chains, and in the decomposition of large systems into smaller, independent subsystems. The block structure allows for efficient storage and computation, making them particularly valuable for large-scale problems.