Maths - Matrix algebra

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, for example, we could have a matrix whose elements are complex numbers.

Relationship to other mathematical quantities

We could think of matrix in other ways, for instance:

• As a vector with an additional dimention.
• As a case of a tensor (rank 2 tensor is a matrix).

Vectors are strongly related to matrices, they can be considered as a one directional matrix, or conversely, we could construct a matrix from a vector (drawn as a column) whose elements are themselves vectors (drawn as a row) :

Strictly I don't think a matrix is a vector of vectors? Because the dot product converts from a vector to a scalar. With matrix multiplication each element is the dot product of a row from one matrix and the column from the other matrix (see matrix arithmetic) Perhaps it would work if the second matrix to be multiplied were transposed first? Can anyone help me work this through.

Regardless of whether we consider vectors as a special case of matrices, or matrices as vectors of vectors, or if we consider vectors and matrices as different types, using vectors and matrices together is very important. A matrix is a way to transform one vector into another vector (and a whole set of vectors into another set of vectors). This allows us to express a linear transform as a single equation:

Another possibility is that matrices can have matrices as elements, provided that the elements are all of the same dimension, when this is the case it can be replaced by one big matrix. For instance, if we have an m x n matrix and each of its elements is a p x q matrix, then we could replace it with a single (m*p) x (n*q) matrix.

For more information about linear transforms using matrices:

• Transforms
  • Multiply matrix by Vector
  • Scaling
  • Translation
  • Rotation

As already mentioned arithmetic is done treating matrices as a unit, the way that this arithmetic is done is explained here:

• Matrix Arithmetic
  • Matrix Addition
  • Matrix Subtraction
  • Identity Matrix
  • Matrix Mutiplication
  • Matrix Division

There are also other functions that can be done on matrices as explained here:

• Matrix Functions
  • Determinant
  • Eigenvalues and eigenvector
  • Trace
• Orthogonal Matrix
  • Rotations
• Affine Transform (4x4 Matrix)
• Matrix Calculus
• Using Matrices in scene graph
• Tutorial
• Code

Matrix Calculus

See here for details

Linear Functions

Standards