Linear Algebra

Linear algebra is a branch of mathematics that deals with vectors, vector spaces, and linear transformations. It is a fundamental tool in engineering and science, providing the mathematical framework for solving systems of linear equations, performing data analysis, and modeling complex phenomena. In SepalSolver, we implement various linear algebra techniques to equip engineers with powerful tools for numerical computation and problem-solving.

Linear algebra techniques are widely used in applications such as computer graphics, machine learning, and structural analysis. They help in optimizing designs, analyzing data, and simulating physical systems.

In this chapter, we will explore different linear algebra methods, their implementations in SepalSolver, and practical examples demonstrating their applications.

• Vectors and Matrices
  • Creating Vectors and Matrices
  • Vectors and Matrices can also be initialized using random
  • Vectors can be initialized using Zeros, Ones, Eye etc
  • Vectors and Matrices can be concatenated
  • Vertical Concatenation
  • Flipping a Matrix
  • Extract a Triangular Portion of Matrix
• Matrix Slicing
  • Extracting/Setting part of a Vector
  • Extracting part of a Matrix
  • Setting Portions of a Matrix
  • Application of Matrix Slicing: Strassen Multiplication
  • Standard vs. Strassen Multiplication
  • Algorithm Steps
  • Advantages
  • Limitations
  • Applications
  • Logical Indexing
    • Complex Conditions
    • Advantages
• Matrix Vectors Operations Null Spaces
  • Arithemetic Operations on Vectors and Matrices
    • Examples
  • Matrix scalar operation
  • Operation with row and column vectors
  • Exploiting Broadcasting (Example)
  • Transpose, Inverse, Determinant, Rref
    • Transpose
    • Inverse
    • Determinant
    • RREF (Reduced Row Echelon Form)
• LU Facorization and FactorUpdate
  • 1. LU Factorization: The Matrix Backbone
  • 2. Rank-1 Updates: Adjusting the Matrix
  • 3. The Sherman-Morrison Formula
  • 4. usage in SepalSolverPython
  • 5. Applications in Industry
• Solution of Linear Systems
  • 1. Direct Methods
  • . 2. Iterative Method
  • . 3. Overdetermined Systems (Least Squares)
  • 3. Matrix Condition Number
• Cholesky Factorization FactorUpdate Positive Definicy
  • Definition
  • Properties
• Generalized Cholesky Factorization and FactorUpdate
• Singular Value Decomposition
• Sparse Matrices
  • Key Characteristics
  • Why Sparse Matrices Matter
  • Applications
  • Common Representations
  • Dynamic Conversion
  • Hybrid Approach
    • Making a SparseMatrix
    • Visualisation
    • Arithmetic Operation
  • LU, iLU, Cholesky and iCholesky Factorization
    • Reodering
• Solution Of Sparse Linear System
• Exercise On Linear Algebra