An example of a matrix elementary functional analysis shilov pdf Jordan normal form. The grey blocks are called Jordan blocks. Jordan matrix, representing the operator with respect to some basis. 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. 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. 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. The Jordan normal form is named after Camille Jordan, who first stated the Jordan decomposition theorem in 1870.