Jordan Matrix - Linear Algebra

Linear Algebra

Any square matrix whose elements are in an algebraically closed field is similar to a Jordan matrix, also in, which is unique up to a permutation of its diagonal blocks themselves. is called the Jordan normal form of 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 .

More generally, given a Jordan matrix, i.e. whose diagonal block, is the Jordan block and whose diagonal elements may not all be distinct, it can easily be seen that the geometric multiplicity of for the matrix, indicated as, corresponds to the number of Jordan blocks whose eigenvalue is . Whereas the index of an eigenvalue for, indicated as, is defined as the dimension of the largest Jordan block associated to that eigenvalue.

The same goes for all the matrices similar to, so can be defined accordingly with respect to the Jordan normal form of for any of its eigenvalues . In this case one can check that the index of for is equal to its multiplicity as a root of the minimal polynomial of (whereas, by definition, its algebraic multiplicity for, is its multiplicity as a root of the characteristic polynomial of, i.e. ). An equivalent necessary and sufficient condition for to be diagonalizable in is that all of its eigenvalues have index equal to, i.e. its minimal polynomial has only simple roots.

Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices): the Jordan decomposition is, in general, a computationally challenging task. From the vector space point of view, the Jordan decomposition is equivalent to finding an orthogonal decomposition (i.e. via direct sums of eigenspaces represented by Jordan blocks) of the domain which the associated generalized eigenvectors make a basis for.