Matrices

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Overview: Matrices

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This article deals with some fundamental matrix features and the basic arithmetic operations.

Matrices can have arbitrary dimensions. A matrix with m rows and n colums is denoted as an m-by-n matrix. In the context of robotics mainly 3-by-3 or 4-by-4 matrices are used. An example of a 3-by-3 matrix is:

$\mathbf{A}=\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13}\\ a_{21} & a_{22} & a_{23}\\ a_{31} & a_{32} & a_{33} \end{array}\right]$

Individual colums and rows are often denoted as column vectors and row vectors. For the example matrix $\mathbf{A}$ the column vectors are

$\left[ \begin{array}{c} a_{11}\\ a_{21}\\ a_{31} \end{array}\right], \left[ \begin{array}{c} a_{12}\\ a_{22}\\ a_{32} \end{array}\right],\text{and } \left[ \begin{array}{c} a_{13}\\ a_{23}\\ a_{33} \end{array}\right]$

and the row vectors are

$\left[ \begin{array}{ccc} a_{11} & a_{12} & a_{13} \end{array}\right], \left[ \begin{array}{ccc} a_{21} & a_{22} & a_{23} \end{array}\right],\text{and } \left[ \begin{array}{ccc} a_{31} & a_{32} & a_{33} \end{array}\right]$

In the following subarticles some basic arithmetic operations for matrices are described.

1. Multiplication with a scalar
2. The transpose of a matrix
3. Addition of matrices
4. Multiplication of matrices
5. Determinant of a matrix
6. Minors and cofactors