Matrices

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`Matrices`

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Matrices[{d₁,d₂}]

represents the domain of matrices of dimensions d₁×d₂.

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Matrices[{d₁,d₂},dom]

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

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Matrices[{d₁,d₂},dom,sym]

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

Details

Valid dimension specifications d_i 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)Summary of the most common use cases

An antisymmetric real matrix in dimension :

In[1]:=1

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https://wolfram.com/xid/0c11hve-3w04hy

Out[1]=1

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

In[2]:=2

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https://wolfram.com/xid/0c11hve-vbw1z4

Out[2]=2

But now it is a symmetric matrix:

In[3]:=3

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https://wolfram.com/xid/0c11hve-ky4r1d

Out[3]=3

Scope (1)Survey of the scope of standard use cases

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

In[1]:=1

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https://wolfram.com/xid/0c11hve-scec08

Out[1]=1

Symmetric real 3×3 matrices:

In[2]:=2

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https://wolfram.com/xid/0c11hve-k0vg4y

Out[2]=2

Antisymmetric matrices:

In[3]:=3

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https://wolfram.com/xid/0c11hve-ifawj8

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0c11hve-59ciq

Out[4]=4

Applications (3)Sample problems that can be solved with this function

Symbolic matrix algebra:

In[1]:=1

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https://wolfram.com/xid/0c11hve-wqz197

In[2]:=2

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https://wolfram.com/xid/0c11hve-ffving

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0c11hve-galrag

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0c11hve-o45zcw

Out[4]=4

Check whether a matrix belongs to a given domain:

In[1]:=1

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https://wolfram.com/xid/0c11hve-2cutil

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0c11hve-f2cq23

Out[2]=2

Conditions involving symbolic parameters may be converted into simpler conditions:

In[3]:=3

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https://wolfram.com/xid/0c11hve-6v1zj9

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0c11hve-rrsgkb

Out[4]=4

In[5]:=5

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https://wolfram.com/xid/0c11hve-1rxsjf

Out[5]=5

Check a subdomain relation:

In[1]:=1

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https://wolfram.com/xid/0c11hve-zamquh

Out[1]=1

Properties & Relations (3)Properties of the function, and connections to other functions

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

In[1]:=1

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https://wolfram.com/xid/0c11hve-bdc6mt

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0c11hve-rejz6b

Out[2]=2

Matrices must be rectangular:

In[1]:=1

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https://wolfram.com/xid/0c11hve-7k9m44

Out[1]=1

Two alternative ways of checking numerical matrices:

In[1]:=1

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https://wolfram.com/xid/0c11hve-epwovp

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0c11hve-pjulfe

Out[2]=2

Possible Issues (2)Common pitfalls and unexpected behavior

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

In[1]:=1

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https://wolfram.com/xid/0c11hve-ljaulo

Out[1]=1

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:

In[1]:=1

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https://wolfram.com/xid/0c11hve-paz34p

Out[1]=1

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