Minimal Polynomial and Jordan Form

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...

Lecture 4: Jordan Canonical Forms

This lecture introduces the Jordan canonical form of a matrix — we prove that every square matrix is equivalent to a (essentially) unique Jordan matrix and we give a method to derive the latter. We also introduce the notion of minimal polynomial and we point out how to obtain it from the Jordan canonical form. Finally, we make an encounter with companion matrices. 1 Jordan form and an applicati...

The cyclic decomposition theorem

8 Matrices 16 8.1 Representation of vector spaces and linear transformations . . . . . . . . . . . . . . . 16 8.2 Similarity transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 8.3 Direct sums of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 8.4 The companion matrix of a polynomial . . . . . . . . . . . . . . . . . . . . . . . ....

The Jordan canonical form

4 The minimal polynomial of a linear transformation 7 4.1 Existence of the minimal polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 The minimal polynomial for algebraically closed fields . . . . . . . . . . . . . . . . . 8 4.3 The characteristic polynomial and the Cayley–Hamilton theorem . . . . . . . . . . . 9 4.4 Finding the minimal polynomial . . . . . . . . . . . . . ....

Minimal Polynomials and Jordan Normal Forms