Algebra III Reduction of Endomorphisms

This course will cover the concepts of symmetric groups and determinants, and will address the reduction of endomorphisms in finite dimensions (up to Jordan form) and its applications. It is a first step toward spectral analysis.

Symmetric group

Notion of a group, group of bijections of X, 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 (link with the volume of parallelograms/parallelepipeds). Determinant of a family of vectors, a matrix, an endomorphism. Cancellation of the determinant. Multiplicativity. Determinant and transposed matrix. Expansion with respect to row or column. Co-matrix and Cramer's formula. Determinant of block matrices.

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

Reduction of endomorphisms:

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

Specific vocabulary: values, vectors, subspaces. Spectrum. Characteristic polynomial.

Endomorphism-diagonalizable matrix-trigonizable. Characterizations by the characteristic polynomial.

Characteristic spaces, lemma of nested kernels, nilpotent endomorphisms.

Polynomials of endomorphisms:

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

Kernel lemma. Characterization of diagonalizable-trigonizable by the minimal polynomial.

Dunford decomposition. Jordan reduction.

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

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

Recommended prerequisites: L1 math

Hourly volumes:

            CM: 30

            TD: 30

            TP:

            Land: