By using a symbol to represent a vector, matrix or array, one gets an efficient notation to model a mathematical problem. Indeed, most scientific, engineering and statistical domains have transitioned to use this type of more abstract and efficient notation. The Wolfram Language has a rich symbolic array language to describe problems. Most high-level solvers support symbolic array expressions and array variables, making it easy and efficient to specify high-dimensional problems.

Symbolic Array Variables

x∈Vectors[…] — assume x is a vector

Matrices   ▪ Arrays

x∈region — assume x is a vector from a geometric region

VectorSymbol — define a vector symbol that can be used together with listable functions

MatrixSymbol   ▪ ArraySymbol   ▪ NonThreadable

Symbolic Array Constants

Common zero-one arrays in array formulas.

SymbolicZerosArray   ▪ SymbolicOnesArray   ▪ SymbolicIdentityArray   ▪ SymbolicDeltaProductArray

Symbolic Array Functions

Dot — vector and matrix inner product

ArrayDot — generalized array inner product

Norm   ▪ Tr   ▪ Det   ▪ Cross   ▪ Transpose   ▪ TensorProduct   ▪ TensorContract   ▪ KroneckerProduct   ▪ TensorWedge

Matrix functions

Inverse   ▪ Adjugate   ▪ PseudoInverse   ▪ LinearSolve   ▪ LeastSquares   ▪ MatrixPower   ▪ MatrixExp   ▪ MatrixLog   ▪ MatrixFunction

Statistics functions

Total   ▪ Mean   ▪ StandardDeviation   ▪ Variance   ▪ Covariance   ▪ Correlation   ▪ AbsoluteCorrelation   ▪ Kurtosis   ▪ Skewness   ▪ Moment   ▪ CentralMoment   ▪ FactorialMoment   ▪ Cumulant

Symbolic Array Predicates

Array equations and inequations

Equal   ▪ Unequal

Array inequalities

VectorLessEqual   ▪ VectorLess   ▪ VectorGreaterEqual   ▪ VectorGreater

Simplification and Transformations

ArraySimplify — simplify symbolic array expressions

ArrayExpand — expand symbolic array expressions

ComponentExpand — expand symbolic array expressions into their components

Array Derivatives

D — symbolic differentiation w.r.t. vector, matrix and array variables

Grad   ▪ Div   ▪ Laplacian

Array Limits

Limit — compute limits using symbolic vector variables

MaxLimit   ▪ MinLimit

Array Algebraic Equation Solvers

Solve — solve equations and inequalities with symbolic array variables

NSolve   ▪ SolveValues   ▪ NSolveValues   ▪ Reduce   ▪ FindInstance   ▪ FindRoot

Array Optimization Solvers

Minimize — optimize using symbolic array variables

MinValue   ▪ ArgMin   ▪ Maximize   ▪ MaxValue   ▪ ArgMax   ▪ NMinimize   ▪ NMinValue   ▪ NArgMin   ▪ NMaximize   ▪ NMaxValue   ▪ NArgMax   ▪ FindMinimum   ▪ FindMinValue   ▪ FindArgMin   ▪ FindMaximum   ▪ FindMaxValue   ▪ FindArgMax

Convex optimization constraints are often expressed using vector and matrix inequalities.

ConvexOptimization   ▪ ParametricConvexOptimization   ▪ RobustConvexOptimization   ▪ LinearOptimization   ▪ LinearFractionalOptimization   ▪ QuadraticOptimization   ▪ SecondOrderConeOptimization   ▪ SemidefiniteOptimization   ▪ GeometricOptimization   ▪ ConicOptimization

Array Integration Solvers

Integrate , NIntegrate — integrate expressions with symbolic vector variables

LineIntegrate   ▪ NLineIntegrate   ▪ SurfaceIntegrate   ▪ NSurfaceIntegrate

Array Differential Equation Solvers

NDSolve — solve differential equations with symbolic array variables

DSolve   ▪ NDSolveValue   ▪ DSolveValue   ▪ ParametricNDSolveValue   ▪ AsymptoticDSolveValue

Array Difference Equation Solvers

RSolve — solve difference equations with symbolic array variables

RSolveValue   ▪ AsymptoticRSolveValue