A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is commonly used in mathematics, science, and engineering to represent and manipulate data, equations, and transformations.
A group is a mathematical concept that consists of a set of elements and a binary operation, such as addition or multiplication, that combines any two elements in the set to produce a third element in the set. Groups have certain properties, such as closure, associativity, identity, and invertibility.
In order to show that matrices form a group, we need to prove that they satisfy the four group properties: closure, associativity, identity, and invertibility. This means that when we perform the binary operation of matrix multiplication on any two matrices, the result will be another matrix within the same set, the operation is associative, there exists an identity matrix within the set, and every matrix has an inverse matrix within the set.
The notation for matrices that form a group is \mathcal{D}_{m',m}^{(j)}, where m' represents the number of rows, m represents the number of columns, and j represents the group number. This notation is commonly used in the study of representation theory, which examines how groups can be represented by matrices.
The concept of matrices forming a group is important because it allows us to use the powerful tools and techniques of group theory to understand and manipulate matrices. This can be particularly useful in areas such as quantum mechanics and physics, where matrices are commonly used to represent physical systems and their transformations.