Mathematical matrix. Matrix multiplication

Even mathematicians of ancient China used in their calculations a record in the form of tables with a certain number of rows and columns. Then similar mathematical objects were called as "magic squares". Although there are known cases of using tables in the form of triangles, which have not been widely used.

To date, a mathematical matrix is understood to mean the volume of a rectangular shape with a given number of columns and symbols, which determine the size of the matrix. In mathematics, this form of writing has found wide application for the recording in a compact form of systems of differential, as well as linear, algebraic equations. It is assumed that the number of rows in the matrix is equal to the number of equations present in the system, the number of columns corresponds to how many unknowns need to be determined in the course of solving the system.

In addition, that the matrix itself in the course of its solution leads to finding unknowns embedded in the condition of the system of equations, there are a number of algebraic operations that can be performed on this mathematical object. This list includes the addition of matrices having the same dimensions. Multiplication of matrices with suitable dimensions (you can multiply only the matrix, on the one hand, having the number of columns equal to the number of rows of the matrix on the other side). It is also possible to multiply the matrix by a vector, or an element of the field or the base ring (otherwise the scalar).

Considering the multiplication of matrices, we must carefully monitor that the number of columns of the first strictly corresponds to the number of rows of the second. Otherwise, this action over the matrices will not be determined. According to the rule by which the matrix is multiplied by a matrix, each element in the new matrix is equated to the sum of the products of the corresponding elements from the rows of the first matrix to elements taken from the columns of the other.

For clarity, consider an example of how matrix multiplication occurs. We take the matrix A

2 3 -2

3 4 0

-1 2 -2,

Multiply it by the matrix B

3 -2

10

4 -3.

The element of the first row of the first column of the resulting matrix is 2 * 3 + 3 * 1 + (-2) * 4. Accordingly, in the first line in the second column there will be an element equal to 2 * (-2) + 3 * 0 + (-2) * (-3), and so on until each element of the new matrix is filled. The rule of matrix multiplication assumes that the result of the product of a matrix with parameters mxn on a matrix having the relation nxk is a table that has dimensions m x k. Following this rule, it can be concluded that the product of so-called square matrices of the same order is always defined.

From the properties that matrix multiplication possesses, it is necessary to single out as one of the main things that this operation is not commutative. That is, the product of the matrix M by N is not equal to the product of N by M. If in square matrices of the same order it is observed that their direct and inverse products are always determined, differing only in the result, then such a definiteness condition is not always satisfied for rectangular matrices.

Multiplication of matrices has a number of properties that have clear mathematical proofs. Associativity of multiplication implies the correctness of the following mathematical expression: (MN) K = M (NK), where M, N, and K are matrices having parameters for which multiplication is defined. The distributivity of multiplication assumes that M (N + K) = MN + MK, (M + N) K = MK + NK, L (MN) = (LM) N + M (LN), where L is a number.

A consequence of the matrix multiplication property, called "associativity", implies that a work containing three or more factors is allowed to record without using brackets.

Using the distributivity property makes it possible to open parentheses when examining matrix expressions. We pay attention, if we open parentheses, then we need to preserve the order of the factors.

The use of matrix expressions allows not only to compactly record cumbersome systems of equations, but also facilitates the process of their processing and solution.