Linear Algebra and Tensor Analysis for Materials Informatics
Linear algebra is an essential mathematical foundation for all fields of materials science, machine learning, and quantum mechanics. In this series, we learn theory and implementation (Python/NumPy/SymPy) in pairs, from vector and matrix basics to eigenvalue problems, singular value decomposition, and tensor algebra. Applications to machine learning (PCA, dimensionality reduction) and materials science (crystallography, elastic tensors) are also covered.
Basic high school mathematics (vector fundamentals) is sufficient. Understanding basic Python usage is desirable. Knowledge of calculus will help deepen understanding.
Learn vector definitions, dot products, cross products, norms, matrix operations (addition, subtraction, multiplication), transpose, inverse matrices, and implement them efficiently with NumPy. Understand the geometric meaning of linear transformations.
Learn determinant definition and properties, Cramer's rule, solution methods for systems of linear equations (Gaussian elimination, LU decomposition), and rank and existence conditions for solutions.
Learn eigenvalue problem definition, characteristic equations, diagonalization, properties of symmetric matrices. Applications to principal component analysis (PCA) and vibration mode analysis in materials science are also covered.
Learn singular value decomposition (SVD) theory, low-rank approximation, applications to image compression and recommendation systems. Understand the relationship between Moore-Penrose pseudo-inverse and least squares method.
Learn tensor definitions and basic operations, tensor products, contraction, symmetric and antisymmetric tensors. Implement applications to stress tensors, strain tensors, elastic tensors, and crystallography.
Upon completing this series, you will achieve:
For more advanced study in this field:
Expand your knowledge with related topics:
Apply your skills to hands-on projects: