# Topics on Matrices

## Objectives

It is intended that students acquire knowledge that is not, generally, object of study in a 1st cycle course of Linear Algebra, with emphasis on a matrix view approach, and which are important not only in their training in this area but also for its applications in other areas (namely in Statistics, Computing, Numerical Analysis and Optimization).

## General characterization

##### Code

12946

##### Credits

6.0

##### Responsible teacher

Carlos Manuel Saiago

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Knowledge corresponding to the contents of a first course of Linear Algebra.

### Bibliography

1. R.A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985 (2nd edition, 2013).

2. C.D. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000.

3. F. Zhang, Matrix Theory - Basic Results and Techniques, Springer, 1999.

4. D.C. Lay, S.R. Lay, J.J. McDonald. Linear Algebra and Its Applications. Pearson Education, 5th edition, 2016.

5. R. Bhatia. Positive Definite Matrices. Princeton University Press, 2007.

### Teaching method

Classes consist on an oral explanation of the theory followed by the resolution of exercises.

### Evaluation method

There are two tests that can substitute the final exam in case of approval. Otherwise the student must succeed the final exam. More detailed rules are available in the Portuguese version.

## Subject matter

0. Review of some basic notions of Linear Algebra

0.1 Matrices

Elementary operations on rows/columns of a matrix. Elementary matrices and equivalence of matrices. Row echelon form and reduced row echelon form/Hermite form. Rank of a matrix.

0.2 Determinants

Definition of determinant and Laplace theorem. Other properties of determinants.

Elementary operations and determinants.

0.3 Linear spaces

Linear spaces and subspaces. Some fundamental subspaces associated to a matrix. Linear independency. Bases and dimension. More about rank of a matrix.

1. Block-partitioned matrices

1.1 Some defitions and operations on block-partitioned matrices

1.2 Some results about block triangular matrices

1.3 Block elementary operations/block elementary matrices

1.4 Rank of the sum and of the product of matrices

1.5 Generalized Laplace theorem and some consequences

1.6 Block elementar transformations and determinants

2. Eigenvalues, eigenvectors and diagonalization

2.1 Eigenvalues and eigenvectors: definition and properties

2.2 Some properties of the characteristic polynomial

2.3 Cayley-Hamilton theorem and some consequences

2.4 Minimal polynomial

2.5 Companion matrices of polynomials

2.6 Similarity and diagonalization

2.7 Simultaneous diagonalization

3. Unitary similarity and triangularization

3.1 Standard inner product on IR^n and on IC^n

3.2 Gram-Schmidt orthogonalization process

3.3 Unitary matrices and orthogonal matrices: definition and characterizations

3.4 Schur decompositiontion theorem and some consequences

3.5 Simultaneous triangularization

4. Normal matrices

4.1 Properties of normal matrices

4.2 Properties of hermitian/skew-hermitian matrices and of symmetric/skew symmetric matrices

4.3 Properties of positive definite /nonnegative definite matrices

4.4 kth root of a positive /nonnegative definite matrix

5. Matrix decompositions

5.1 Full rank decomposition

5.2 Singular value decomposition

6. Generalized inverses

6.1 The Moore-Penrose inverse