What are matrices?
The matrices are a two-dimensional set of numbers or symbols distributed in a rectangular shape in vertical and horizontal lines so that their elements are arranged in rows and columns. They are useful for describing systems of linear or differential equations, as well as representing a linear application.
Every matrix is represented by a capital letter, and its elements are given in lowercase letters in a list enclosed by parentheses or square brackets. Each, in turn, has a double superscript: the first refers to the row and the second to the column to which it belongs.
This mathematical expression can be added, multiplied, and decomposed, so it is commonly used in linear algebra.
What concepts are associated with matrices?
Some of the concepts needed to complete the definition and analysis of matrices are:
- Elements: the numbers that make up the matrix.
- Dimension: the result of the number of rows times the number of columns. The letter m is used to designate the number of rows and n for the number of columns.
- Rings: this algebra term refers to the system formed by a set of internal operations that respond to a set of properties. Matrices are understood as elements of a ring.
- Function: a correspondence rule between two sets in which an element of the first set corresponds exclusively to a single element of the second set.
What types of matrices are there?
A matrix can be:
- Rectangular: it has different numbers of rows and columns.
- Row: a rectangular array with a single row.
- Column: a rectangular matrix with a single column.
- Null: an array that has zero elements.
- Square of order n: a matrix that has the same number of rows as columns. In this type of matrix, the dimension is called the order, and its value coincides with the number of rows and columns.
- Diagonal: a kind of square matrix where the elements not located on the main diagonal are equal to zero.
- Scalar: a diagonal matrix where all the elements on the main diagonal are equal.
- Identity: this is a scalar matrix where the elements of the main diagonal are equal to one, while all other elements are equal to zero.
- Inverse: the opposite of another matrix whose elements have signs opposite to the main matrix. That is, the inverse matrix of A is called -A, and all the elements of the set are the opposites of the elements of matrix A.
- Transpose: the matrix obtained when converting rows to columns. The superscript t is used to represent it, and its dimension is n x m.
- Upper triangular: this is a square matrix where at least one of the terms above the main diagonal is non-zero, and all those below the main diagonal are equal to zero.
- Lower triangular: unlike the previous type, this type of matrix has at least one element below the main diagonal that is non-zero, and all those above the main diagonal are equal to zero.
How can matrices be used?
Matrices have multiple applications, especially for representing coefficients in systems of equations or linear applications; a matrix can perform the same function as vector data in a linear system of application. Depending on this, some applications include:
- In computer science: one of the fields where matrices are most used, given their effectiveness in working with information. Matrices are ideal for graphic representations and animating shapes.
- In robotics: matrices are used for programming robots that can execute different tasks. One example of this is a bionic arm that can use programmable mechanical processes to fulfill functions similar to those of a human arm. All of this programming is the result of calculations using matrices.