Virtual Book > Mathematics and Algorithms > Linear Algebra in Mathematica
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Additional functionality related to this tutorial has been introduced in subsequent versions of Mathematica. For the latest information, see Matrices and Linear Algebra.

Linear Algebra in Mathematica

IntroductionIntroduction

Tensors and Arrays

Matrices as Mathematica Expressions

Expression Input and Output

Design Principles of Mathematica

Matrix and Tensor OperationsMatrix and Tensor Operations

Building MatricesBuilding Matrices

Special Matrices

Structural OperationsStructural Operations

Getting Pieces of Matrices

Getting Multiple Pieces

Setting Pieces of Matrices

Setting Multiple Pieces

Extracting Submatrices

Deleting Rows and Columns

Inserting Rows and Columns

Extending Matrices

Transpose

Rotating Elements

Testing Matrices

Further Structural Operations

Element-wise OperationsElement-wise Operations

Listability

Map

Vectors and TensorsVectors and Tensors

Testing Vectors and Tensors

Visualization of MatricesVisualization of Matrices

Formatting Matrices

Plotting Matrices

Import and Export of Matrices

Matrix MultiplicationMatrix Multiplication

Outer Product

Visualization of the Outer Product

Generalized Inner Product

Matrix PermutationsMatrix Permutations

Permutation Matrices

Working with Sparse ArraysWorking with Sparse Arrays

Basic OperationsBasic Operations

SparseArray

Rule Inputs for SparseArray

Banded Sparse Matrices

Identity and Diagonal Sparse Matrices

Normal

ArrayRules

Structural OperationsStructural Operations

Getting Pieces of Matrices

Getting Multiple Pieces

Setting Pieces of Matrices

Setting Multiple Pieces

Extracting Submatrices

Deleting Rows and Columns

Inserting Rows and Columns

Extending Matrices

Transpose

Rotating Elements

Testing Matrices

Further Structural Operations

Element-wise OperationsElement-wise Operations

Listability

Map

Visualization of Sparse MatricesVisualization of Sparse Matrices

Formatting Sparse Matrices

Plotting Sparse Matrices

Import and Export of Sparse Matrices

Matrix MultiplicationMatrix Multiplication

Outer Product

Matrix Permutations

Converting Equations to Sparse Arrays

SparseArray Data Format

Matrix ComputationsMatrix Computations

Basic OperationsBasic Operations

NormsNorms

Vector Norms
Matrix Norms

NullSpace

Rank

Reduced Row Echelon Form

Inverse

PseudoInverse

Determinant

Minors

Solving Linear SystemsSolving Linear Systems

Singular Matrices

Homogeneous Equations

Estimating and Calculating Accuracy

Symbolic and Exact Matrices

Row Reduction

Saving the Factorization

Methods

LAPACK

Multifrontal

Krylov

Cholesky

Symbolic Methods

Least Squares SolutionsLeast Squares Solutions

Data Fitting

Eigensystem ComputationsEigensystem Computations

Eigensystem Properties

Diagonalizing a Matrix

Symbolic and Exact Matrices

Generalized Eigenvalues

Methods

LAPACK

Arnoldi

Symbolic Methods

Matrix DecompositionsMatrix Decompositions

LU Decomposition

Cholesky Decomposition

Cholesky and LU Factorizations

Orthogonalization

QR Decomposition

Solving Systems of Equations

Singular Value Decomposition

Generalized Singular Values

Options

Schur Decomposition

Generalized Schur Decomposition

Options

Jordan Decomposition

Functions of Matrices

Matrix TypesMatrix Types

Numbers in MathematicaNumbers in Mathematica

Exact versus Approximate Numbers

Mixed Mode Arithmetic

Matrices in MathematicaMatrices in Mathematica

Standard Numerical Techniques

Arbitrary-Precision Numerical Techniques

Symbolic Algebra Techniques

Mixed Mode Matrices

Complex Matrices

Arbitrary-Precision Matrices

Performance of Linear Algebra ComputationPerformance of Linear Algebra Computation

Packed ArraysPacked Arrays

Packed Array Functions

Packed Array Operations

Packed Array Summary

Programming EfficiencyProgramming Efficiency

Measuring Performance

Vectorizing Loops

List Creation

List Updating

Using Built-in Support

A Slow Way

A Faster Way

Also Fast but Neater

Matrix Contents

Mixed Symbolic/Numerical Matrices

Mixed Numerical Type Matrices

Integer Matrices

Expression EfficiencyExpression Efficiency

Updating of Matrices

Appending to Matrices

Linear Algebra ExamplesLinear Algebra Examples

Matrix Ordering

Full Rank Least Squares SolutionsFull Rank Least Squares Solutions

Least Squares Cholesky

Least Squares QR

Minimization of 1 and Infinity NormsMinimization of 1 and Infinity Norms

One-Norm Minimization

Infinity-Norm Minimization

Finite Difference Solutions

Mesh PartitioningMesh Partitioning

Data

Plotting the Mesh

The Laplacian

The Fiedler Vector

Partitioning the Nodes

Matrix Functions with NDSolve

ReferencesReferences

Software ReferencesSoftware References

ARPACK

ATLAS

Harwell-Boeing

Matrix Market

METIS

TAUCS

UMFPACK

Other References

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