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. It is a first step towards spectral analysis.

Symmetrical group

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

Determinants:

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

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

Reduction of endomorphisms:

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

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

Diagonalizable-trigonalizable matrix-endomorphism. Characterizations by the characteristic polynomial.

Characteristic spaces, nested kernel lemma, nilpotent endomorphisms.

Polynomials of endomorphisms:

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

Lemma of kernels. Characterization of diagonalizable-trigonalizable by the minimal polynomial.

Dunford's decomposition. Jordan reduction.

Applications: computation of 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 the complex plane

Recommended prerequisites: L1 math

Hourly volumes:

            CM : 30

            TD : 30

            TP:

            Terrain: