Computational Aspects of the Jordan Canonical Form

Infinite-dimensional versions of the primary, cyclic and Jordan decompositions

The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

NUROP Congress Paper Jordan Canonical Forms of Linear Operators

Any linear transformation can be represented by its matrix representation. In an ideal situation, all linear operators can be represented by a diagonal matrix. However, in the real world, there exist many linear operators that are not diagonalizable. This gives rise to the need for developing a system to provide a beautiful matrix representation for a linear operator that is not diagonalizable....

Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A ∈ Cm×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA∗ and A∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AA and AA. More generally, we consider the matrix triple (A,G, Ĝ), where G ∈ Cm×m, Ĝ ∈ Cn×n ar...

Determination of a Matrix Function in the Form of f(A)=g(q(A)) Where g(x) Is a Transcendental Function and q(x) Is a Polynomial Function of Large Degree Using the Minimal Polynomial

Matrix functions are used in many areas of linear algebra and arise in numerical applications in science and engineering. In this paper, we introduce an effective approach for determining matrix function f(A)=g(q(A)) of a square matrix A, where q is a polynomial function from a degree of m and also function g can be a transcendental function. Computing a matrix function f(A) will be time- consu...

Jordan Canonical Form: Theory and Practice