Introduction to Matrix Groups

Now that you have an idea of what a group is and what some if its properties are and what some of its properties are, we can move on to a specific kind of group. A collection of matrices can form a group in the same way that a collection of individual numbers can form a group. A Matrix Group is just a group with some other special qualities* and is defined as follows:

Matrix Group
A collection of square matrices that satisfies the group properties. The group composition is matrix multiplication and the group inverse is the matrix inverse.

Click on green terms in the definition to get a further explanation.

A square matrix is any matrix with the same number of rows and columns. It can also be called an n x n matrix.

Example of matrix multiplication:

\left[ \matrix{a & c \\ b & d} \right] \left[ \matrix{A & C \\ B & D} \right]

The equation shows a 2x2 matrix multiplied with a 2x2 matrix. This will yield a 2x2 matrix, because there are two rows in the first matrix and 2 columns in the second matrix. If the equation was a 3x2 matrix multiplied by a 2x2, the result would be 3x2.

So the solution looks like $$\left[\matrix{x_{1,1} & x_{1,2} \\ x_{2,1} & x_{2,2}}\right]$$. If we label the rows and columns of the first matrix as $$r\left(i\right),c\left(i\right)$$ and those of the second matrix as $$R\left(i\right),C\left(i\right)$$, then we can define the solutions as follows:

\begin{eqnarray} x_{1,1} = r\left(1\right) \cdot C\left(1\right) = a \times A + c \times B \\ x_{1,2} = r\left(1\right) \cdot C\left(2\right) = a \times C + c \times D \\ x_{2,1} = r\left(2\right) \cdot C\left(1\right) = b \times A + d \times B \\ x_{2,2} = r\left(2\right) \cdot C\left(2\right) = b \times C + d \times D \end{eqnarray}

Note: only certain dimension matrices can be multiplied together. The first matrix must have the same number of columns as the second matrix has rows.

The matrix inverse of a matrix A is the matrix $$A^{-1}$$ such that $$AA^{-1}=1$$ where 1 is the identity matrix of appropriate dimension. There are many ways to find the matrix inverse, the most common method being Gauss-Jordan elimination. There is also a formula for 2x2 matrices that is easy to use.