# Jordan Normal Form

In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator on some basis. The form is characterized by the condition that any non-diagonal entries that are non-zero must be equal to 1, be immediately above the main diagonal (on the superdiagonal), and have identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis on which the matrix has the required form exists if and only if all eigenvalues of M lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.

If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size. The Jordanâ€“Chevalley decomposition is particularly simple on a basis on which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.

The Jordan normal form is named after Camille Jordan.

Read more about Jordan Normal Form:  Motivation, Complex Matrices, Real Matrices, Consequences, Example, Numerical Analysis, Powers

### Famous quotes containing the words jordan, normal and/or form:

We do not deride the fears of prospering white America. A nation of violence and private property has every reason to dread the violated and the deprived.
—June Jordan (b. 1939)

A normal adolescent is so restless and twitchy and awkward that he can mange to injure his knee—not playing soccer, not playing football—but by falling off his chair in the middle of French class.
Judith Viorst (20th century)

I do not mean to imply that the good old days were perfect. But the institutions and structure—the web—of society needed reform, not demolition. To have cut the institutional and community strands without replacing them with new ones proved to be a form of abuse to one generation and to the next. For so many Americans, the tragedy was not in dreaming that life could be better; the tragedy was that the dreaming ended.
Richard Louv (20th century)