Linear Algebra II


The student is supposed to consolidate and complement knowledge acquired in Linear Algebra I (vide syllabus). In learning process, logical reasoning and critical mind should continue being developed.

General characterization





Responsible teacher

Maria de Fátima Vale de Gato Santos Rodrigues


Weekly - 6

Total - 84

Teaching language



Knowledge corresponding to the contents of Linear Algebra I (1st semester-1st year)

Knowledge about inner product spaces. The necessary notions and results constitute the first part of the syllabus of the unit Geometry.


1. Apostol, T. M., Linear Algebra – a first course with applications to differential equations, John Wiley & Sons, 1997.

2. Anton, H., e Rorres, C., Elementary Linear Algebra -  Applications Version, 9th Edition, John Wiley & Sons, 2005.

3. Friedberg, S.H., Insel, A. J., e Spence, L. E., Linear Algebra, 3rd Ed., Prentice Hall, 1997.

4. Horn, R. A., e Johnson, C. R., Matrix Analysis, Cambridge University Press, 1985.

5. Leon, S. J., Linear Algebra with Applications, 7th Ed., Prentice Hall, 2006.

6. A. P. Santana, J. F. Queiró, Introdução à Álgebra Linear, Gradiva, 2010.

Teaching method

There are classes in which theory is lectured and illustrated by examples. There are also problem-solving sessions. For each chapter there is a list of proposed exercises that the students should solve. Most of the exercises is corrected in the problem-solving sessions.

Evaluation method

Students must attend at least 2/3 of the problem-solving classes.

The students that do not fulfill the above requirements automatically fail "Álgebra Linear II". 

There are two mid-term tests. These tests can substitute the final exam if the student has grade, at least, 7.5 in the second one and CT is, at least, 9.5.  CT is calculated as follows:

 CT = 0,50*T1 + 0,50*T2 where Ti, 1 ≤ i ≤ 2, is the non-rounded grade obtained in test i.

To be approved in final exam, the student must have a minimum grade of 9.5 in it.

More detailed rules are available in the portuguese version. 

The non-portuguese students are advised to address the professor for more detailed information.

Subject matter

1. Eigenvalues and eigenvectors of endomorphisms and matrices – Definitions and properties. Eigenspaces. Characteristic polynomial. Algebraic and geometric multiplicities. Diagonalization. Cayley-Hamilton theorem. Minimum polynomial.

 2. Endomorphisms of inner product spaces (finite dimension) – Adjoint of an endomorphism; normal endomorphism, hermitian (symmetric), skew-hermitian (skew-symmetric), unitary (orthogonal) and respective definitions for square matrices. Positive definite endomorphism, positive semidefinite, negative definite, negative semidefinite, indefinite and respective definitions for square matrices. Relationship between the different types of endomorphisms and the respective matrices determined by an orthonormal basis. Fundamental results involving these notions, in particular, Schur theorem and spectral theorem.

 3. Jordan canonical form and some of its fundamental consequences.


Programs where the course is taught: