Numerical Linear Algebra

Notes will be updated here periodically.

This page is last updated on Feb 20, 2026.

Pre-requisites: 

MA-II (MA002), Programming

Course Syllabus: 

[Unit-1] Introduction: Recall Matrix Algebra, computing determinants, trace, inverse, adjoint, transpose and their computational cost; Various types of matrices and their properties. Elementary and permutation matrices, elementary transformation, row echelon, and reduced row echelon forms. Rank, properties of rank, applications of rank. Matrix Norms, inner products, condition number. Vector space, subspaces, basis, dimension, null-space, rank-nullity theorem. Linear transformations, similar matrices.   

[Unit-2] Eigen-vales and Eigen-vectors: Eigenvalues and eigenvectors, characteristic equations, minimal polynomial. Positive and negative definite matrices. Gershgorin theorem. Diagonalizability of matrices. Power method to find the largest and smallest eigen values. Given – Householder method. LR and QR method. Singular value decomposition (SVD).  Eigenvalues of large sparse matrices.    

[Unit-3] System of Linear Equations - Direct Methods: Recall system of linear equations, Direct and Indirect methods, Cramer’s rule, Gauss Jorden, Gauss Elimination, pivoting – partial and complete. LU decomposition – Crout, Doolittle and Cholesky. Banded linear systems; tri-diagonal systems. Structured linear systems.  

[Unit-4] System of Linear Equations - Indirect Methods: Indirect methods – Gauss Jacobi, Gauss Seidel, Successive Over Relaxation (SOR) method and their convergence criterion. Preconditioning, Krylov subspace method. Methods to deal with sparse matrices.  

Course Outcomes: 

The present course will provide a computational perspective of the linear algebra and its implementation. By attending the course, students will be able to understand 

1. Computing various operations related to matrices and associated computational costs.

2. Linear spaces, subspaces, linear transformations, and related concepts.

3. Eigenvalues and eigenvalues, various methods, and applications.

4. Direct and indirect methods to solve system of linear equations.       

Recommended Books:

Text Books:

1. Lloyd N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM (1997).

2. Biswa Nath Dutta, Numerical Linear Algebra and Applications, SIAM, 2010.

3. Roger A. Horn and Charles R. Johnson, Matrix Analysis, Cambridge University Press, 1994. 

4. William Ford, Numerical Linear Algebra with Applications, Academic Press, 2014.  

Reference Books:

1. Watkins, David S. Fundamentals of matrix computations. Vol. 64. John Wiley & Sons, 2004.

2. H A V. Sundarapandian, Numerical Linear Algebra, PHI, 2008.

Evaluation Scheme: 

Students will be evaluated on a scale of 150 ={MTE, ETE, Practical, Project} where

MTE - Open Book Mid Term Exam - 30 Marks for 2 Hr duration

ETE - Open Book End Term Exam - 50 Marks for 2 Hr duration

Practical - MATLAB Coding Exam of 30 Marks for 1 Hr duration

Project - Project (a group of max 4) of 40 Marks 

Note: A minimum of 35 out of 150 marks is required to pass the course. The grading will be relative-scaled.

Suggested Project Topics: 

1. Matrix decomposition (all possible matrix decomposition, their requirement and applications)

2. Multi-grid (geometric and algebraic) methods to solve system of linear equations

3. Graph neural network (GNN) to solve system of linear equations 

4. Least square problems (various methods and applications)

5. Principal component analysis (PCA) and its applications

6. Parallel matrix computations

7. Use of optimization techniques in matrix computations

8. Computational random matrix theory

9. Secured matrix computations (use of cryptography and steganography)

10. Computational methods for large sparse matrices

(one can choose other topics as well after prior approval)

Note: (1) A single project can be taken by atmost 02 groups. Each group can have atmost 4 members. Single member is also allowed.

(2) Project marks (40) = Report in IEEE format (20M) + Presentation (20M). 

(3) The last date to finalize the project topic and group members is 25 January 2026. 

Exam/Test Schedule: 

1. Mid Term Exam (MTE) will be held on 23rd Feb 2026 from 10:00 to 11:30. 

2. The last date to submit the abstract of the project (200-300 words with 5-10 reference articles) is 9th March 2026.