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 Jordan–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:
“All of Western tradition, from the late bloom of the British Empire right through the early doom of Vietnam, dictates that you do something spectacular and irreversible whenever you find yourself in or whenever you impose yourself upon a wholly unfamiliar situation belonging to somebody else. Frequently its your soul or your honor or your manhood, or democracy itself, at stake.”
—June Jordan (b. 1939)
“Every normal person, in fact, is only normal on the average. His ego approximates to that of the psychotic in some part or other and to a greater or lesser extent.”
—Sigmund Freud (18561939)
“Every New Englander might easily raise all his own breadstuffs in this land of rye and Indian corn, and not depend on distant and fluctuating markets for them. Yet so far are we from simplicity and independence that, in Concord, fresh and sweet meal is rarely sold in the shops, and hominy and corn in a still coarser form are hardly used by any.”
—Henry David Thoreau (18171862)