Algebra III: Reduction of Endomorphisms

This course will cover the concepts of symmetric groups and determinants, and will address the reduction of endomorphisms to finite dimensions (up to Jordan form) and its applications. It serves as an introduction to spectral analysis.

Symmetric group

The concept of a group, the group of bijections from X, the group S_n. Decomposition into a product of cycles with disjoint supports. Order of a permutation. Transpositions and signature morphisms.

Determinants:

Alternating n-linear form (connection to the volume of parallelograms/parallelepipeds). Determinant of a family of vectors, a matrix, or an endomorphism. Zero determinant. Multiplicativity. Determinant and transpose of a matrix. Row- or column-major expansion. Co-matrix and Cramer’s rule. Determinants of block matrices.

A reinterpretation of the Gauss pivot algorithm: the matrices (I+E_ij) and the permutations generate GL(E). Calculation of the determinant using the Gauss pivot method.

Reduction of endomorphisms:

Review: change of basis and transition matrix, direct sums of vector subspaces, stable subspaces, and block diagonal matrices.

Relevant vocabulary: eigenvalues, eigenvectors, subspaces. Spectrum. Characteristic polynomial.

Endomorphism—diagonalizable matrix—trigonally diagonalizable matrix. Characterizations via the characteristic polynomial.

Characteristic spaces, nested kernel lemma, nilpotent endomorphisms.

Endomorphism polynomials:

Evaluation morphism. Minimal polynomial of an endomorphism. Cayley-Hamilton theorem (for example, via companion matrices).

Kernel lemmas. Characterization of diagonalizable-trigonizable functions using the minimal polynomial.

Dunford decomposition. Jordan reduction.

Applications: calculation of matrix powers, linear recurrence sequences, and systems of homogeneous linear differential equations.

L1 Linear Algebra (HAX102X and HAX202X) and HAX104X – Geometry in the Plane and the Complex Plane

Recommended prerequisites: First-year math

Hourly volumes:

            CM: 30

            TD: 30

            Practical Work:

            Lot: