A matrix is a rectangular array of elements which are operated on as a single object. The elements are often numbers but could be any mathematical object provided that it can be added and multiplied with acceptable properties (it must be a field structure - see box on right for details), for example, we can have a matrix whose elements are complex numbers.
Applications of Matrices
Matrices are widely used in geometry, physics and computer graphics applications. For instance this page discusses how they can be used to represent transforms.
Relationship to other mathematical quantities
We could think of matrix in many ways, for instance:
- As a vector with rows and columns.
- As a case of a tensor (rank 2 tensor is a matrix).
- As a representation of a set of linear equations.
Matrices are a vector space, that is they have the following operations:
| operation |
notation |
explanation |
| addition |
M[a+b] = M[a] + M[b] |
the addition of two matrices is done by adding the corresponding elements of the two matrices. |
| scalar multiplication |
M[s*a] = s * M[a] |
a scalar product of a matrices is done by multiplying the scalar product with each of its terms individually. |
In addition to being a vector space, matricies have additional structure defined by another multiplication type, this allows us to multiply two matrices and get a third matrix (provided certain conditions about the dimensions are met). This type of multiplication can be thought of as an extension to the 'dot' product of vectors, it is the main type of matrix multiplication and it is discussed on this page.
Axioms
| axiom |
addition |
scalar multiplication |
multiplication |
| associativity |
(a+b)+c=a+(b+c) |
(s1 s2) a = s1 (s2 a) |
(a*b)*c=a*(b*c) |
| commutativity |
a+b=b+a |
not necessarily commutative |
| distributivity |
s*(b+c)=s*b+s*c
(s1+s2)*a=s1*a+s2*a |
a*(b+c)=a*b+a*c
(a+b)*c=a*c+b*c |
| identity |
a+0 = a
0+a = a |
1*a = a |
a*1 = a
1*a = a |
| inverses |
a+(-a) = 0
(-a)+a = 0 |
a*a-1 = 1
a-1*a = 1
if det[a]≠0 |
This table gives some of the properties of the matrix operations, note that matrix multiplication is not commutative.
Relationship to Vectors
Vectors are strongly related to matrices, they can be considered as a one directional matrix. So can we create a matrix from a vector whose elements are themselves vectors? We cannot do this because the elements of the vector must be a mathematical structure known as a 'field' and a vector is not itself a field because it does not necessarily have commutative multiplication and other properties required for a field.
Still it would be nice if we could construct a matrix from a vector (drawn as a column) whose elements are themselves vectors (drawn as a row) :