Form of a matrix indicating its eigenvalues and their algebraic multiplicities
Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number lambda on its main diagonal, and ones above the main diagonal. The lambdas are the eigenvalues of the matrix; they need not be distinct.
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),[1][2]
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them.
Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix 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 algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.[3][4][5]
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.[3][4][5]
The Jordan–Chevalley decomposition is particularly simple with respect to a basis for 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.[6][7][8]
The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.[9]
^
Shilov defines the term Jordan canonical form and in a footnote says that Jordan normal form is synonymous.
These terms are sometimes shortened to Jordan form. (Shilov)
The term Classical canonical form is also sometimes used in the sense of this article. (James & James, 1976)
^Holt & Rumynin (2009, p. 9)
^ abBeauregard & Fraleigh (1973, pp. 310–316)
^ abGolub & Van Loan (1996, p. 355)
^ abNering (1970, pp. 118–127)
^Beauregard & Fraleigh (1973, pp. 270–274)
^Golub & Van Loan (1996, p. 353)
^Nering (1970, pp. 113–118)
^Brechenmacher, "Histoire du théorème de Jordan de la décomposition matricielle (1870-1930). Formes de représentation et méthodes de décomposition", Thesis, 2007
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