# Matrices

Matrices[{d1,d2}]

represents the domain of matrices of dimensions d1×d2.

Matrices[{d1,d2},dom]

represents the domain of matrices of dimensions d1×d2, with components in the domain dom.

Matrices[{d1,d2},dom,sym]

represents the subdomain of matrices d1×d2 with symmetry sym.

# Details

• Valid dimension specifications di in Matrices[{d1,d2},dom,sym] are positive integers. It is also possible to work with symbolic dimension specifications.
• Valid component domain specifications dom are either Reals or Complexes. Matrices[{d1,d2}] uses Complexes by default.
• For matrices, the symmetry sym can be either Symmetric[{1,2}], Antisymmetric[{1,2}], or {}, which represents the trivial symmetry.
• If the symmetry sym is nontrivial, then the dimensions d1 and d2 must coincide.

# Examples

open allclose all

## Basic Examples(1)

An antisymmetric real matrix in dimension :

The Dot product of with itself is also a × matrix:

But now it is a symmetric matrix:

## Scope(1)

Declare matrices of any dimensions, with complex entries and no symmetry:

Symmetric real 3×3 matrices:

Antisymmetric matrices:

## Applications(3)

Symbolic matrix algebra:

Check whether a matrix belongs to a given domain:

Conditions involving symbolic parameters may be converted into simpler conditions:

Check a subdomain relation:

## Properties & Relations(3)

Matrices can also be defined using Arrays with rank 2. These two assumptions are equivalent:

Matrices must be rectangular:

Two alternative ways of checking numerical matrices:

## Possible Issues(2)

Addition of symbolic and explicit matrices is determined by the Listable attribute of Plus:

Hence, listability will in general affect operations that simultaneously involve both symbolic and explicit matrices.

The zero matrix may be represented as 0 in symbolic computations:

Wolfram Research (2012), Matrices, Wolfram Language function, https://reference.wolfram.com/language/ref/Matrices.html.

#### Text

Wolfram Research (2012), Matrices, Wolfram Language function, https://reference.wolfram.com/language/ref/Matrices.html.

#### CMS

Wolfram Language. 2012. "Matrices." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Matrices.html.

#### APA

Wolfram Language. (2012). Matrices. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Matrices.html

#### BibTeX

@misc{reference.wolfram_2024_matrices, author="Wolfram Research", title="{Matrices}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/Matrices.html}", note=[Accessed: 19-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_matrices, organization={Wolfram Research}, title={Matrices}, year={2012}, url={https://reference.wolfram.com/language/ref/Matrices.html}, note=[Accessed: 19-July-2024 ]}