- Docente: Alessandro D'Andrea
- Credits: 6
- SSD: MAT/03
- Language: Italian
- Moduli: Alessandro D'Andrea (Modulo 1) Nicoletta Cantarini (Modulo 2)
- Teaching Mode: Traditional lectures (Modulo 1) Traditional lectures (Modulo 2)
- Campus: Bologna
- Corso: First cycle degree programme (L) in Mathematics (cod. 6061)
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from Feb 18, 2025 to May 29, 2025
Learning outcomes
Successful students will acquire a broad understanding of linear algebra notions such as canonical forms of endomorphisms.
They will be able to apply such knowledge to elementary geometric problems.
Course contents
Quotient vector spaces. Homomorphism and isomorphism theorem.
Decomposition in generalized eigenspaces. Jordan normal form. Invariants of the Jordan form and Jordan form of companion matrices. Cayley-Hamilton theorem.
Primary decomposition and primary components. Rational normal form. Similarity of matrices is field independent.
Bilinear forms: symmetric, skew-symmetric, (non)degenerate. Real bilinear matrices: positive and negative (semi)definite ones. Signature.
Complex sesquilinear matrices: hermitian, skew-hermitian, positive definite forms. Complex scalar products.
(Right and left) kernel of a bilinear form. Dimension of (right and left) orthocomplement of a subspace. Orthogonal decompositions.
Matrices associated to bilinear forms. Congruence of matrices.
Diagonalization of symmetrich bilinear forms. Decomposition of alternating forms into orthogonal sum of their kernel and symplectic planes.
Complex spectral theorem for normal operators. Real spectral theorem for self-adjoint operatore. Spectral theorem for real symmetric bilinear forms.
Readings/Bibliography
S. Lang "Linear Algebra"
Teaching methods
Frontal lectures at the blackboard.
Resolution of exercises in recitation sections.
Assessment methods
Written and oral exams.
Teaching tools
Recitation sections.
Office hours
See the website of Alessandro D'Andrea
See the website of Nicoletta Cantarini