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

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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:

Introduced in 2012
 (9.0)