Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Numerical Analysis → Numerical Linear Algebra ↓


Matrix Factorizations


Matrix factorization is an important concept in numerical linear algebra, an important area of numerical analysis. This topic is relevant in many applications in science, engineering, and mathematics. The idea is to express a given matrix as a product of two or more matrices with specific properties. This can make solving linear systems, finding inverses, and determining matrix properties computationally efficient. In this lesson, we will explore different types of matrix factorization, their purposes, and simple examples of their applications.

Basic terminology and concepts

Before diving into specific matrix factorizations, it is important to understand the basic matrix operations and properties. Matrices are two-dimensional arrays of numbers that can be added, subtracted, and multiplied by following certain rules. A fundamental operation is the multiplication of matrices, which is non-commutative, meaning that multiplying a matrix A by B does not always give the same result as multiplying B by A

Matrix factorization involves decomposing a matrix into products of simpler matrices. It can help solve the following types of equations:

Ax = b

where A is a matrix, and x and b are vectors.

LU decomposition

LU decomposition is a method in which a matrix A is decomposed into two matrices, L and U L is a lower triangular matrix, and U is an upper triangular matrix. This factorization is possible if A is a square matrix, which means it has the same number of rows and columns.

The goal of LU decomposition is to simplify the process of solving linear systems and calculating determinants and inverses. A matrix is decomposed as follows:

A = LU

Here's a simple example with a 3x3 matrix:

A = | 2 3 1 |
| 4 7 -1 |
| -2 4 5 |
L = | 1 0 0 |
| 2 1 0 |
| -1 1 1 |
U = | 2 3 1 |
| 0 1 -3 |
| 0 0 7 |

Through forward and backward substitution on L and U, solving Ax = b becomes more computationally feasible.

l You LU decomposition

Cholesky decomposition

The Cholesky decomposition is a specific type of LU decomposition that applies to Hermitian, positive-definite matrices. It expresses the matrix A as:

A = LL T

where L is a lower triangular matrix with real and positive diagonal entries, and L T is the transpose of L

Consider a symmetric matrix example:

A = | 4 12 -16 |
| 12 37 -43 |
| -16 -43 98 |
L = | 2 0 0 |
| 6 1 0 |
| -8 5 3 |
L T = | 2 6 -8 |
| 0 1 5 |
| 0 0 3 |

Applications in system solutions

The Cholesky decomposition is useful for solving the system Ax = b where A is symmetric and positive definite. It is advantageous because of its efficiency and numerical stability.

l L T Cholesky decomposition

QR decomposition

The QR decomposition decomposes a matrix A into the product of two matrices Q and R, where Q is an orthogonal matrix, and R is an upper triangular matrix:

A = QR

It is particularly useful in solving linear systems and eigenvalue calculations, and can also be used in calculating the least squares solution of Ax = b.

Consider this example:

A = | 1 2 |
| 3 4 |
| 5 6 |
Q = | 1/√35 2/√35 |
| 3/√35 4/√35 |
| 5/√35 6/√35 |
R = | √35 0 |
| 0 √35 |
Why R QR decomposition

Singular value decomposition (SVD)

One of the most powerful decomposition techniques, singular value decomposition, represents the matrix A as follows:

A = UΣV *

where U and V are unitary matrices, and Σ (sigma) is a diagonal matrix.

SVD is particularly important because it can be applied to any mxn matrix, not just square matrices. It is used in signal processing, statistics, and computer vision.

Example:

A = | 1 0 1 |
| 0 1 0 |
| 1 0 1 |
U = | 1/√2 0 1/√2 |
| 0 1 0 |
| 1/√2 0 1/√2 |
Σ = | 2 0 0 |
| 0 1 0 |
| 0 0 0 |
V * = | 1/√2 0 1/√2 |
| 0 1 0 |
| -1/√2 0 1/√2 |

Applications of SVD

SVD is used in solving systems of equations, signal processing, and statistics. Its application in Principal Component Analysis (PCA) helps reduce the dimensionality of the dataset, which is important for machine learning applications.

You Σ V * SVD

Practical considerations

Matrix factorizations are not just theoretical constructs but are implemented in various computational libraries and software such as LAPACK, MATLAB and NumPy in Python. These libraries help in performing efficient numerical calculations required for large-scale problems.

When applying decompositions, numerical stability and computational cost are important considerations. Depending on the properties of the matrix (e.g., sparse, symmetric, etc.), certain decompositions are more suitable.

Conclusion

Matrix factorizations such as LU, Cholesky, QR, and SVD are integral tools in numerical linear algebra. They streamline operations on matrices, such as solving linear systems and analyzing matrix properties, making them invaluable for practical applications. Understanding these factorizations expands your ability to solve complex computational problems efficiently and accurately.


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