Course

Unit 1

Advanced Linear Algebra Foundations

• Normed and inner product spaces
• Orthogonality and projections
• Matrix norms and operator norms
• Positive definite and semidefinite matrices
• Kronecker products and block matrices
• Sparse and structured matrices

Research Relevance: Feature spaces, kernel methods, high-dimensional representations

Unit 2

Numerical Linear Algebra and Matrix Computations

• Conditioning and numerical stability
• Direct and iterative methods for linear systems
• Krylov subspace methods (CG, GMRES)
• Randomized numerical linear algebra
• Low-rank matrix approximation

Research Relevance: Large-scale data analytics, recommender systems

Unit 3

Eigenvalue Problems and Matrix Factorizations

• Eigenvalue algorithms and spectral theory
• Singular Value Decomposition (SVD)
• Generalized eigenvalue problems
• Non-negative Matrix Factorization (NMF)
• Tensor decompositions (CP, Tucker)

Research Relevance: Dimensionality reduction, spectral clustering, graph analytics

Unit 4

Convex Optimization Theory

• Convex sets and convex functions
• First and second-order optimality conditions
• Lagrangian duality and KKT conditions
• Sensitivity and perturbation analysis
• Interior-point methods

Research Relevance: Convex relaxations, sparse optimization

Unit 5

Large-Scale and Non-Convex Optimization

• Gradient descent and accelerated methods
• Stochastic optimization (SGD, mini-batch methods)
• Proximal and coordinate descent methods
• Optimization in deep learning
• Distributed and parallel optimization

Research Relevance: Deep learning, scalable AI systems

Text Books / References

• Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press.

Additional References

• Golub, G. H. and Van Loan, C. F., Matrix Computations
• Trefethen, L. N. and Bau, D., Numerical Linear Algebra
• Nocedal, J. and Wright, S., Numerical Optimization

Course Objective

To develop rigorous theoretical and algorithmic foundations in applied linear algebra and optimization, enabling PhD scholars to model, analyze, and solve large-scale computational problems arising in advanced computer science research.

Course Outcomes (COs)

• CO1:Analyze advanced vector spaces, matrix structures, and spectral properties used in computational models.
• CO2:Design and analyze numerical linear algebra algorithms for large-scale and high-dimensional data.
• CO3:Apply eigenvalue-based methods and matrix factorizations in machine learning and data analysis.
• CO4:Formulate and solve convex optimization problems with rigorous theoretical guarantees.
• CO5:Apply large-scale and non-convex optimization techniques to contemporary computer science research problems.

Evaluation Pattern

CO–PO Mapping Matrix