Matrix addition
Introduction
Matrix addition is an operation in linear algebra that takes a pair of matrices, and produces another matrix. Numbers such as the real or complex numbers can be added according to elementary arithmetic. On the other hand, matrices are arrays of numbers, so there is a need to define operations like addition and multiplication in a way that extends these basic operations.
Definition
Matrix addition is defined element-wise, that is, the elements of the resulting matrix are the sums of the corresponding elements of the original matrices. This means that in order to add two matrices, they must have the same dimensions. More formally, if A and B are both m × n matrices, then their sum, C = A + B, is also an m × n matrix, and its elements are defined by the equation c_ij = a_ij + b_ij for all i and j.
Properties
Matrix addition has a number of properties which make it a well-behaved mathematical operation. These properties are often used in proofs and computations involving matrices.
1. Commutativity: A + B = B + A for all matrices A and B of the same size. 2. Associativity: (A + B) + C = A + (B + C) for all matrices A, B, and C of the same size. 3. Identity element: There exists an m × n matrix 0 (the zero matrix) such that A + 0 = A for all m × n matrices A. 4. Inverse element: For every m × n matrix A, there exists an m × n matrix -A (the negative of A) such that A + (-A) = 0. 5. Distributivity over scalar multiplication: a(A + B) = aA + aB for all real numbers a and all m × n matrices A and B. 6. Distributivity over matrix multiplication: A(B + C) = AB + AC and (B + C)A = BA + CA for all matrices A, B, and C for which the operations are defined.
Matrix Addition and Linear Systems
Matrix addition plays a crucial role in the study of systems of linear equations. The system of equations can be written in matrix form, and then solved using matrix operations, including addition. This approach is particularly useful for systems with many variables and equations, where manual computation becomes cumbersome.
Matrix Addition in Computer Science
In computer science, matrix addition is a fundamental operation in many algorithms and data structures. For example, in graph theory, matrices are used to represent graphs, and matrix addition can be used to combine graphs or to find paths within a graph. In image processing, matrices are used to represent images, and matrix addition can be used to blend images or to apply filters.