Jordan normal form

This type of structure is required to describe the Jordan normal form.

This example shows how to calculate the Jordan normal form of a given matrix.

In this case A is similar to a matrix in Jordan normal form.

While the Jordan normal form determines the minimal polynomial, the converse is not true.

There is a standard form for the consimilarity class, analogous to the Jordan normal form.

This form is sometimes called the generalized Jordan normal form, or primary rational canonical form.

The Jordan normal form is named after Camille Jordan.

In a different direction, for compact operators on a Banach space, a result analogous to the Jordan normal form holds.

The simpler triangularization result is often sufficient however, and in any case used in proving the Jordan normal form theorem.

Matrices in Jordan normal form.

For non-diagonalizable matrices one can calculate the Jordan normal form followed by a series expansion, similar to the approach described in logarithm of a matrix.

For example, row echelon form and Jordan normal form are canonical forms for matrices.

The Jordan-Chevalley decomposition is particularly simple on a basis on which the operator takes its Jordan normal form.

This is the case for modules over a field or PID, and underlies Jordan normal form of operators.

Jordan normal form of a matrix is a complete invariant for matrices up to conjugation, but eigenvalues (with multiplicities) are not.

In linear algebra, a Jordan normal form (often called Jordan canonical form)

This is the Jordan normal form of A. The section Example below fills in the details of the computation.

It can be shown that the Jordan normal form of a given matrix A is unique up to the order of the Jordan blocks.

For this reason, the Jordan normal form is usually avoided in numerical analysis; the stable Schur decomposition is often a better alternative.

Any normal matrix is similar to a diagonal matrix, since its Jordan normal form is diagonal.

Knowing the algebraic and geometric multiplicities of the eigenvalues is not sufficient to determine the Jordan normal form of 'A'.

The Jordan normal form of reads , where is an upper diagonal matrix containing the eigenvalues and ; hence, .

In a sentence, the qualitative behaviour of such a dynamical system may substantially change as the versal deformation of the Jordan normal form of .

The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.

All matrices, whether they are diagonalizable or not, have a Jordan normal form , where the matrix J consists of Jordan blocks.