Ali Hassan Farhadi: En studie om minimalpolynom av linjära operatorer och Jordans normalform

Abstract.

Linear algebra is used in various fields, especially in computational data processing. Linear transformations which is central to the subject, allow problems to be solved using matrices and vectors. This thesis explores linear transformations through minimal polynomials and Sheldon Axler’s theorem which claims that eigenvalues are more naturally treated without determinants. We discuss minimal polynomials, their relation to eigenvalues, eigenvectors, characteristic polynomials, and Jordans normalform. The thesis begins with key concepts from basic linear algebra, including eigenvalues, eigenvectors, and diagonalization. It introduces generalized eigenvectors to study non-diagonalizable operators and proves eigenvalues exist without using determinants. The main focus is on minimal polynomials, including the Cayley–Hamilton theorem, which relates the characteristic polynomial to an annihilating polynomial. We prove the uniqueness of minimal polynomials and their connection to Jordans normalform. The thesis also compares eigenvalue treatments with and without determinants and offering an alternative method for finding eigenvalues. Finally, Jordans normalform is used in solving ordinary differential equations.