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Concept
Jordan matrix
Summary
In the mathematical discipline of matrix theory, a Jordan matrix, named after Camille Jordan, is a block diagonal matrix over a ring R (whose identities are the zero 0 and one 1), where each block along the diagonal, called a Jordan block, has the following form:
Every Jordan block is specified by its dimension n and its eigenvalue , and is denoted as Jλ,n. It is an matrix of zeroes everywhere except for the diagonal, which is filled with and for the superdiagonal, which is composed of ones.
Any block diagonal matrix whose blocks are Jordan blocks is called a Jordan matrix. This (n1 + ⋯ + nr) × (n1 + ⋯ + nr) square matrix, consisting of r diagonal blocks, can be compactly indicated as or , where the i-th Jordan block is Jλi,ni.
For example, the matrix
is a 10 × 10 Jordan matrix with a 3 × 3 block with eigenvalue 0, two 2 × 2 blocks with eigenvalue the imaginary unit i, and a 3 × 3 block with eigenvalue 7. Its Jordan-block structure is written as either or diag(J0,3, Ji,2, Ji,2, J7,3).
Any n × n square matrix A whose elements are in an algebraically closed field K is similar to a Jordan matrix J, also in , which is unique up to a permutation of its diagonal blocks themselves. J is called the Jordan normal form of A and corresponds to a generalization of the diagonalization procedure. A diagonalizable matrix is similar, in fact, to a special case of Jordan matrix: the matrix whose blocks are all 1 × 1.
More generally, given a Jordan matrix , that is, whose kth diagonal block, , is the Jordan block Jλk,mk and whose diagonal elements may not all be distinct, the geometric multiplicity of for the matrix J, indicated as , corresponds to the number of Jordan blocks whose eigenvalue is λ. Whereas the index of an eigenvalue for J, indicated as , is defined as the dimension of the largest Jordan block associated to that eigenvalue.
The same goes for all the matrices A similar to J, so can be defined accordingly with respect to the Jordan normal form of A for any of its eigenvalues .
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