Matrix Theory

Contents

1. Modeling using matrices: Basic examples, graph theory (adjacency/Laplacian), probability and
statistics (Covariance matrices, Markov matrices), signal processing (Fourier matrices,
convolution/filtering), communication (Linear codes, multiple antenna communications), machine learning (clustering,
dimensionality reduction), basic linear dynamical systems etc
2. Row/column spaces and rank, basic matrix operations (multiplication, transpose, determinant,
trace, inverse, etc), matrix types, rank-nullity, underdetermined/overdetermined systems of
linear equations, block matrices
3. Matrix decompositions - eigenvalues, SVD and applications like PCA, LU and Cholesky
decomposition, QR and Schur decomposition, non-negative matrix factorization, Quadratic
forms
4. More on determinants - its algebraic properties, how to compute determinants
5. Characteristic polynomial and its properties
6. Solving Ax=b, least squares and its many variants, min norm solutions for underdetermined
systems of linear equations, other optimization problems framed using matrices; pseudo
inverse, matrix norms
7. Special matrices: Toeplitz, Circulant, Fourier etc.
8. Generalized eigenvectors and Jordan form
9. Solutions to systems of ordinary differential equations, matrix exponent
10. Numerical issues and common matrix algorithms, linear algebra in MATLAB/python
11. Introduction to random matrices and context in which they are useful.

References

1.G. Strang. Introduction to Linear Algebra, Wellesley Cambridge Press.
2.R. A. Horn and C. R. Johnson Matrix Analysis, Cambridge University Press.
3.. Boyd and L. Vandenberghe, Introduction to Applied Linear Algebra Vectors, Matrices,
and Least Squares, Cambridge University Press.