Jordannormálforma

Jordannormálforma, often abbreviated as JNF, is a canonical form for a linear transformation on a finite-dimensional vector space over an algebraically closed field. It is a specific representation of a matrix that reveals a great deal about the underlying linear transformation, particularly its eigenvalues and generalized eigenvectors. The Jordannormálforma is obtained by a similarity transformation, meaning it is related to the original matrix by an invertible matrix.

The Jordannormálforma consists of a block diagonal matrix where each block is a Jordan block. A Jordan block is a square matrix with a single eigenvalue on the diagonal, ones on the superdiagonal (immediately above the main diagonal), and zeros everywhere else. The size of the Jordan blocks and the eigenvalues on their diagonals are uniquely determined by the linear transformation.

The existence and uniqueness of the Jordannormálforma are fundamental results in linear algebra. It is particularly useful when the characteristic polynomial of a linear transformation can be factored completely over the field, which is guaranteed if the field is algebraically closed, such as the complex numbers. If a linear transformation is diagonalizable, its Jordannormálforma is a diagonal matrix, which is a special case of the Jordannormálforma where all Jordan blocks are of size 1x1.

The Jordannormálforma plays a crucial role in the study of differential equations, control theory, and various areas of applied mathematics and physics, providing a simplified way to analyze the behavior of systems described by linear transformations. It allows for a systematic understanding of the structure of a vector space under the action of a linear operator.