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