blockdiagonalization

Blockdiagonalization is the process of transforming a linear operator or matrix into a block diagonal form, by finding a basis in which the matrix acts independently on a direct sum of invariant subspaces. Concretely, a linear transformation T on a finite-dimensional vector space V is block diagonalizable if there exists a decomposition V = U1 ⊕ U2 ⊕ ... ⊕ Uk into T-invariant subspaces such that, in a basis adapted to this decomposition, the matrix of T has the form diag(T|U1, T|U2, ..., T|Uk). Equivalently, there are commuting projections onto the Ui whose images are precisely the invariant subspaces.

A standard route to block diagonalization uses the primary decomposition: if the minimal polynomial of T factors into pairwise coprime components, then V splits into invariant subspaces corresponding to these factors, and T becomes block diagonal with blocks given by the restrictions to each subspace. In many practical applications, one seeks a decomposition into minimal invariant blocks that render the operator easier to study, for example by simplifying exponentials e^{tA} or decoupling linear systems.

Blockdiagonalization is stronger than merely triangularization; while every matrix admits an upper triangular form, not every matrix can be reduced to block diagonal form with nontrivial blocks. The field and the availability of invariant subspaces influence feasibility. Related concepts include the rational and Jordan canonical forms, which describe canonical block structures, and the broader idea of simultaneous block diagonalization for families of commuting operators.

Applications include decoupling linear dynamical systems, simplifying spectral analysis, and enabling parallel computation by treating blocks independently.