blockdiagonalizations

Block diagonalization refers to the process of transforming a square matrix into a block diagonal form by a similarity transformation. Specifically, a matrix A in n×n has a block diagonalization if there exists an invertible P such that P^{-1} A P = diag(A1, A2, ..., Ak), where each Ai is square and corresponds to an invariant subspace Vi of the underlying vector space, with V = ⊕ Vi. The blocks capture the action of A on each invariant subspace.

Existence and theory: The decomposition is possible exactly when the space can be written as a direct sum of A-invariant subspaces. In finite dimensions, this is guaranteed when the subspaces are kernels of coprime factors of the minimal polynomial, via the primary decomposition. Over an algebraically closed field, A is similar to a block diagonal matrix with blocks that are the Jordan blocks associated with eigenvalues grouped into blocks. The primary decomposition yields V = ⊕ ker p_i(A) for coprime p_i, providing a natural block form.

Methods: Practical construction uses invariant subspaces found via eigenvectors or generalized eigenvectors, or via families of commuting matrices that share invariant subspaces. Permutation similarity can reorder the basis to reveal blocks. Block diagonal forms are closely connected to canonical forms such as Jordan and rational canonical forms, which organize the action of A on invariant subspaces into standard blocks.

Applications: Block diagonalization simplifies the computation of powers and matrix exponentials, helps solve linear systems, and decouples dynamics in control theory, differential equations, and Markov chains. By reducing a problem to smaller, independent subsystems, it often reduces computational complexity and clarifies the structure of the transformation or system.