This book is about the topic of mathematical analysis, particularly in the field of engineering. It attempts to be a companion piece to high-level engineering texts that will rely on a certain fundamental mathematical background among readers.

This will build on topics covered in Probability, Algebra, Linear Algebra, Calculus, and Ordinary Differential Equations. Readers of this book are expected to have background knowledge in all those topics. Topics covered will be inter-disciplinary engineering topics, and will be highly mathematical. However, overlap between this book and other mathematics books, except where necessary, will be minimized.

This book is intended to accompany a graduate course of study in engineering analysis.

• Vector Spaces
• Vector Basics
• Linear Independence and Basis
• Linear Transformations
• Minimization
• Projections
• Linear Spaces
• Matrices
• Matrix Forms
• Quadratic Forms
• Eigenvalues and Eigenvectors
• Diagonalization
• Spectral Decomposition
• Error Estimation

• Matrix Functions
• Cayley-Hamilton Theorem
• Matrix Exponentials
• Lyapunov Equation
• Function Spaces
• L₂ Space
• Banach and Hilbert Spaces
• Fourier Series
• Arbitrary Basis Expansion
• Bessel Equation and Parseval Theorem
• Multi-Dimensional Fourier Series

• Wavelets
• Wavelet Transforms

• Random Variables
• Probability Functions
• Distributions
• Expectation and Entropy
• SISO Transformations
• MISO Transformations
• Correlation
• Random Vectors

• Leibnitz' Rule
• Differential Operator Theory
  • Linear Operators
• Complex Analysis

• Optimization
• Unconstrained Optimizations
• Constrained Optimizations

• Resources