blockdiagonalize

Blockdiagonalize refers to the process of transforming a linear operator or matrix into a block diagonal form, via a suitable change of basis. If A is an n-by-n matrix representing a linear transformation on a vector space V, blockdiagonalization seeks a basis of V in which the matrix of the transformation is block diagonal: P^{-1}AP = diag(A1, A2, ..., Ak), where each Ai is a square matrix acting on a subspace Vi and V decomposes as a direct sum V = V1 ⊕ V2 ⊕ ... ⊕ Vk of A-invariant subspaces Vi.

The key condition is the existence of a decomposition of V into A-invariant subspaces that are complementary. In such a basis, the action of A preserves each Vi and the matrix representation has zero blocks off the diagonal. The sizes of the blocks correspond to the dimensions of the invariant subspaces. Blockdiagonalization thus reduces a problem about A to smaller, independent problems on the blocks Ai.

Common theoretical tools include the primary decomposition and invariant subspace theory. If the minimal polynomial factors into coprime polynomials, V decomposes into a direct sum of kernels associated with those factors, yielding a natural block diagonal form. In practice, blockdiagonalization is achieved by finding invariant subspaces (for example, spanned by eigenvectors or generalized eigenvectors) and choosing a basis adapted to this decomposition. Not every matrix is block diagonalizable with real or complex entries; for instance, an indecomposable Jordan block cannot be further blockdiagonalized, though matrices with distinct eigenvalues are diagonalizable, which is a special case of blockdiagonalization.