Algebra III Reduction of endomorphisms

This course will cover the notions of symmetric group, determinants and will deal with the reduction of endomorphisms in finite dimension (up to Jordan form) and its applications. This is a first step towards spectral analysis.

Symmetrical group

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

Determinants:

Alternating n-linear form (link with volume of parallelograms/parallelepipeds). Determinant of a family of vectors, a matrix, an endomorphism. Determinant cancellation. Multiplicativity. Determinant and transpose matrix. Development with respect to row or column. Co-matrix and Cramer formula. Block matrix determinant.

Re-interpretation of the Gauss pivot algorithm: matrices (I+E_ij) and permutations generate GL(E). Calculate the determinant by Gauss pivot.

Reduction of endomorphisms:

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

Vocabulary: values, vectors, subspaces. Spectrum. Characteristic polynomial.

Diagonalizable-trigonalizable matrix-endomorphism. Characterization by characteristic polynomial.

Characteristic spaces, nested kernel lemma, nilpotent endomorphisms.

Polynomials of endomorphisms:

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

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

Dunford decomposition. Jordan reduction.

Applications: calculating powers of a matrix, linear recurrent sequences, systems of homogeneous linear differential equations.

L1 linear algebra (HAX102X and HAX202X ) and HAX104X - Geometry in the plane and complex plane

Recommended prerequisites: L1 maths

Hourly volumes :

            CM: 30

            TD : 30

            TP :

            Terrain :