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Understanding Non-Abelian Matrices in Matrix Multiplication
Understanding Non-Abelian Matrices in Matrix Multiplication
Sometimes, the question "How are abstract algebra matrices non-abelian?" can be interpreted as when two matrices A and B do not commute under matrix multiplication, i.e., AB ≠ BA. Being abelian or commuting is a property of pairs of elements under a binary operation, but in the case of matrices, this property is far from universal.
General Properties of Matrix Multiplication
Firstly, for two matrices A and B to even have a defined product, they need to meet certain dimension requirements. Specifically, if A is a p × q matrix and B is a q × r matrix, then AB is defined, but BA might not be defined if p ≠ r. This is a key reason why not all pairs of matrices commute.
However, when we restrict ourselves to p × p square matrices, both AB and BA are defined, and this leads us to explore more about the non-abelian property. In the square matrix case, while the identity matrix multiples commute with every matrix, the vast majority of matrices do not commute with other matrices. This is because, for square matrices, there exist matrices that do not commute with even the most basic matrices.
Non-Abelian Semigroups and Groups
The set of all p × p square matrices forms a non-abelian semigroup with the identity matrix serving as the identity element under multiplication. Among these, the set of all non-singular (invertible) p × p matrices forms a non-abelian group. Within these groups, subgroups can be formed where all matrices commute with each other. These subgroups are termed as abelian groups
Another crucial point is that the terms abelian and non-abelian are most commonly associated with algebraic structures such as groups, rings, and fields, rather than matrices alone. For instance, non-abelian under multiplication would be more precise than stating abstract algebra matrices are non-abelian. The property of being abelian or non-abelian is a fundamental characteristic of the entire algebraic structure, not just of matrices.
Examples and Verification
To illustrate why not all matrices commute, consider two simple 2 × 2 matrices:
A begin{bmatrix} 1 2 3 4 end{bmatrix}, quad B begin{bmatrix} 0 1 1 0 end{bmatrix}
Calculating the matrix products AB and BA, we get:
AB begin{bmatrix} 1 2 3 4 end{bmatrix} begin{bmatrix} 0 1 1 0 end{bmatrix} begin{bmatrix} 2 1 4 3 end{bmatrix}, quad BA begin{bmatrix} 0 1 1 0 end{bmatrix} begin{bmatrix} 1 2 3 4 end{bmatrix} begin{bmatrix} 3 4 1 2 end{bmatrix}
Clearly, AB ≠ BA, which demonstrates the non-commutative nature of matrix multiplication for these two matrices.
In conclusion, while not all pairs of matrices are non-abelian under multiplication, the non-commutative nature of matrix multiplication is a fundamental property of many matrix pairs. This is why it is important to understand the conditions under which matrix multiplication commutes and the implications of non-commutativity.