Jordankanonisk

Jordankanonisk refers to the Jordan canonical form of a square matrix, a standard representation up to similarity transformations. For a matrix A, there exists an invertible P such that P^{-1} A P equals a block-diagonal matrix J, where each block is a Jordan block associated with an eigenvalue of A.

A Jordan block J_k(λ) is a k-by-k matrix with λ on the diagonal, ones on the superdiagonal, and zeros elsewhere. The full Jordankanonisk is composed of such blocks, grouped by eigenvalue. The eigenvalues are the roots of the characteristic polynomial of A, and the sizes and numbers of blocks reflect the structure of the eigenvectors and generalized eigenvectors (the chains).

Key properties include existence and uniqueness up to permutation of blocks when working over an algebraically closed field (typically the complex numbers). The Jordankanonisk is diagonal precisely when all blocks are 1x1, i.e., when A is diagonalizable. If not, some blocks have size greater than one, capturing the presence of generalized eigenvectors. Over the reals, a real Jordan form may include 2x2 blocks corresponding to complex conjugate pairs of eigenvalues.

Computation involves finding eigenvalues, constructing generalized eigenvector chains for each eigenvalue, and assembling a basis that yields the similarity transform P. In practice, numerical methods may favor the Schur form for robustness; real Jordan form can also be represented with 2x2 blocks for complex pairs.

Historically, the Jordankanonisk originated from Camille Jordan’s work in the 19th century on classifying linear operators. It provides insight into the geometric and algebraic multiplicities of eigenvalues and underpins methods for computing functions of matrices and solving linear differential equations.