Jordanmuoto

Jordanmuoto (Finnish: literally "Jordan form") is a term used to refer to the Jordan canonical form, a classification of square matrices or linear operators up to similarity. It expresses a linear operator as a block diagonal matrix whose blocks, called Jordan blocks, each correspond to an eigenvalue and have that eigenvalue on the diagonal with ones on the superdiagonal.

A Jordan block J_k(λ) is a k×k matrix with λ on every diagonal entry and 1 on each entry immediately above the diagonal; all other entries are zero. A matrix is similar to a direct sum of such Jordan blocks when the underlying field is algebraically closed (for example, the complex numbers). Over non-algebraically closed fields, a real canonical form uses 2×2 blocks to represent complex conjugate eigenvalue pairs.

The Jordan form is unique up to the ordering of the Jordan blocks. Structural information is reflected in block sizes: the size of the largest Jordan block for an eigenvalue equals the exponent of that eigenvalue in the minimal polynomial, and the number of blocks equals the geometric multiplicity of the eigenvalue. Jordan chains of generalized eigenvectors provide the change-of-basis that produces the canonical form.

Jordan form has applications in solving linear differential equations, computing matrix functions (notably the exponential), and classifying linear operators by similarity. In numerical practice, direct computation can be unstable, so alternative decompositions such as the Schur form are often preferred.