Matrix size m ? n is a rectangular table of numbers containing m rows and n columns. The numbers that make up the matrix are called elements matrices.
Matrices are denoted by capital letters of the Latin alphabet ( A,B,C...), and to designate matrix elements, lowercase letters with double indexing are used:
Where i- line number, j- column number.
For example, matrix
Or in shorthand, A=(); i=1,2…, m; j=1,2, …, n.
Other matrix notations are used, for example: , ? ?.
Two matrices A And IN same size are called equal, if they coincide element by element, i.e. = , where i= 1, 2, 3, …, m, A j= 1, 2, 3, …, n.
Let's consider the main types of matrices:
1. Let m = n, then matrix A is a square matrix that has order n:
The elements form the main diagonal, the elements form the secondary diagonal.
The square matrix is called diagonal, if all its elements, except perhaps the elements of the main diagonal, are equal to zero:
A diagonal, and therefore square, matrix is called single, if all elements of the main diagonal are equal to 1:
Note that the identity matrix is the matrix analogue of one in the set of real numbers, and we also emphasize that the identity matrix is defined only for square matrices.
Here are examples of identity matrices:
Square matrices
are called upper and lower triangular, respectively.
- 2. Let m= 1, then the matrix A- row matrix, which looks like:
- 3. Let n=1, then the matrix A- column matrix, which looks like:
4. A zero matrix is a matrix of order mn, all elements of which are equal to 0:
Note that the null matrix can be a square matrix, a row matrix, or a column matrix. The zero matrix is the matrix analogue of zero in the set of real numbers.
5. A matrix is called transposed to a matrix and is denoted if its columns are the rows of the matrix corresponding in number.
Example. Let
Note that if the matrix A has order mn, then the transposed matrix has the order nm.
6. Matrix A is called symmetric if A =, and skew-symmetric if A =.
Example. Examine for matrix symmetry A And IN.
hence the matrix A- symmetrical, because A =.
hence the matrix IN- skew-symmetrical, since B = -.
Note that symmetric and skew-symmetric matrices are always square. Any elements can be on the main diagonal of a symmetric matrix, and identical elements must be placed symmetrically relative to the main diagonal, that is, the main diagonal of a skew-symmetric matrix always contains zeros, and symmetrically relative to the main diagonal
matrix square laplace cancellation
Note that matrix elements can be not only numbers. Let's imagine that you are describing the books that are on your bookshelf. Let your shelf be in order and all books be in strictly defined places. The table, which will contain a description of your library (by shelves and the order of books on the shelf), will also be a matrix. But such a matrix will not be numeric. Another example. Instead of numbers there are different functions, united by some dependence. The resulting table will also be called a matrix. In other words, a Matrix is any rectangular table made up of homogeneous elements. Here and further we will talk about matrices made up of numbers.
Instead of parentheses, square brackets or straight double vertical lines are used to write matrices
 |
(2.1*) |
Definition 2. If in the expression(1) m = n, then they talk about square matrix, and if , then oh rectangular.
Depending on the values of m and n, some special types of matrices are distinguished:
The most important characteristic square matrix is her determinant or determinant, which is made up of matrix elements and is denoted
Obviously, D E =1; .
Definition 3. If , then the matrix A called non-degenerate or not special.
Definition 4. If detA = 0 , then the matrix A called degenerate or special.
Definition 5. Two matrices A And B are called equal and write A = B if they have the same dimensions and their corresponding elements are equal, i.e..
For example, the matrices and are equal, because they are equal in size and each element of one matrix is equal to the corresponding element of the other matrix. But the matrices cannot be called equal, although the determinants of both matrices are equal, and the sizes of the matrices are the same, but not all elements located in the same places are equal. The matrices are different because they have different sizes. The first matrix is 2x3 in size, and the second is 3x2. Although the number of elements is the same - 6 and the elements themselves are the same 1, 2, 3, 4, 5, 6, but they are in different places in each matrix. But the matrices are equal, according to Definition 5.
Definition 6. If you fix a certain number of matrix columns A and the same number of rows, then the elements at the intersection of the indicated columns and rows form a square matrix n- th order, the determinant of which called minor k – th order matrix A.
Example. Write down three second-order minors of the matrix
Various operations are performed on such matrices: they multiply by each other, find determinants, etc. Matrix- a special case of an array: if an array can have any number of dimensions, then only a two-dimensional array is called a matrix.
In programming, a matrix is also called a two-dimensional array. Any of the arrays in the program has a name, as if it were a single variable. To clarify which of the array cells is meant, when it is mentioned in the program, the number of the cell in it is used together with the variable. Both a two-dimensional matrix and an n-dimensional array in a program can contain not only numeric, but also symbolic, string, Boolean and other information, but always the same within the entire array.
Matrices are denoted by capital letters A:MxN, where A is the name of the matrix, M is the number of rows in the matrix, and N is the number of columns. Elements are represented by corresponding lowercase letters with indices indicating their number in the row and column a (m, n).
The most common matrices are rectangular in shape, although in the distant past mathematicians also considered triangular ones. If the number of rows and columns of a matrix is the same, it is called square. In this case, M=N already has the name of the matrix order. A matrix with only one row is called a row. A matrix with only one column is called a columnar matrix. A diagonal matrix is a square matrix in which only the elements located along the diagonal are non-zero. If all elements are equal to one, the matrix is called identity; if all elements are equal to zero, it is called zero.
If you swap rows and columns in a matrix, it becomes transposed. If all elements are replaced by complex conjugates, it becomes complex conjugate. In addition, there are other types of matrices, determined by the conditions that are imposed on the matrix elements. But most of these conditions apply only to square ones.
Video on the topic
Matrices in mathematics are one of the most important objects of practical importance. Often an excursion into the theory of matrices begins with the words: “A matrix is a rectangular table...”. We will start this excursion from a slightly different direction.
Phone books of any size and with any amount of subscriber data are nothing more than matrices. Such matrices look approximately like this:
It is clear that we all use such matrices almost every day. These matrices come with a different number of rows (they vary like a directory issued by a telephone company, which can have thousands, hundreds of thousands and even millions of lines, and a new notebook you just started, which has less than ten lines) and columns (a directory of officials of some kind). some organization in which there may be columns such as position and office number and your same address book, where there may not be any data except the name, and thus there are only two columns in it - name and telephone number).
All sorts of matrices can be added and multiplied, as well as other operations can be performed on them, but there is no need to add and multiply telephone directories, there is no benefit from this, and besides, you can use your mind.
But many matrices can and should be added and multiplied and thus solve various pressing problems. Below are examples of such matrices.
Matrices in which the columns are the production of units of a particular type of product, and the rows are the years in which the production of this product is recorded:
You can add matrices of this type, which take into account the output of similar products by different enterprises, in order to obtain summary data for the industry.
Or matrices consisting, for example, of one column, in which the rows are the average cost of a particular type of product:
The last two types of matrices can be multiplied, and the result is a row matrix containing the cost of all types of products by year.
Matrices, basic definitions
A rectangular table consisting of numbers arranged in m lines and n columns is called mn-matrix (or simply matrix ) and is written like this:
(1)
In matrix (1) the numbers are called its elements (as in the determinant, the first index means the number of the row, the second – the column at the intersection of which the element is located; i = 1, 2, ..., m; j = 1, 2, n).
The matrix is called rectangular , If .
If m = n, then the matrix is called square , and the number n is its in order .
Determinant of a square matrix A is a determinant whose elements are the elements of a matrix A. It is indicated by the symbol | A|.
The square matrix is called not special (or non-degenerate , non-singular ), if its determinant is not zero, and special (or degenerate , singular ) if its determinant is zero.
The matrices are called equal , if they have the same number of rows and columns and all corresponding elements match.
The matrix is called null , if all its elements are equal to zero. We will denote the zero matrix by the symbol 0 or .
For example,
Matrix-row (or lowercase ) is called 1 n-matrix, and matrix-column (or columnar ) – m 1-matrix.
Matrix A", which is obtained from the matrix A swapping rows and columns in it is called transposed relative to the matrix A. Thus, for matrix (1) the transposed matrix is
Matrix transition operation A" transposed with respect to the matrix A, is called matrix transposition A. For mn-matrix transposed is nm-matrix.
The matrix transposed with respect to the matrix is A, that is
(A")" = A .
Example 1. Find matrix A" , transposed with respect to the matrix
and find out whether the determinants of the original and transposed matrices are equal.
Main diagonal A square matrix is an imaginary line connecting its elements, for which both indices are the same. These elements are called diagonal .
A square matrix in which all elements off the main diagonal are equal to zero is called diagonal . Not all diagonal elements of a diagonal matrix are necessarily nonzero. Among them there may be equal to zero.
A square matrix in which the elements on the main diagonal are equal to the same number, non-zero, and all others are equal to zero, is called scalar matrix .
Identity matrix is called a diagonal matrix in which all diagonal elements are equal to one. For example, the third-order identity matrix is the matrix
Example 2. Given matrices:
Solution. Let us calculate the determinants of these matrices. Using the triangle rule, we find
Matrix determinant B let's calculate using the formula 
We easily get that 
Therefore, the matrices A and are non-singular (non-degenerate, non-singular), and the matrix B– special (degenerate, singular).
The determinant of the identity matrix of any order is obviously equal to one.
Solve the matrix problem yourself, and then look at the solution
Example 3. Given matrices
,
,
Determine which of them are non-singular (non-degenerate, non-singular).
Application of matrices in mathematical and economic modeling
Structured data about a particular object is simply and conveniently recorded in the form of matrices. Matrix models are created not only to store this structured data, but also to solve various problems with this data using linear algebra.
Thus, a well-known matrix model of the economy is the input-output model, introduced by the American economist of Russian origin Vasily Leontiev. This model is based on the assumption that the entire production sector of the economy is divided into n clean industries. Each industry produces only one type of product, and different industries produce different products. Due to this division of labor between industries, there are inter-industry connections, the meaning of which is that part of the production of each industry is transferred to other industries as a production resource.
Product volume i-th industry (measured by a specific unit of measurement), which was produced during the reporting period, is denoted by and is called full output i-th industry. Issues can be conveniently placed in n-component row of the matrix.
Number of units i-industry that needs to be spent j-industry for the production of a unit of its output is designated and called the direct cost coefficient.
1st year, higher mathematics, studying matrices and basic actions on them. Here we systematize the basic operations that can be performed with matrices. Where to start getting acquainted with matrices? Of course, from the simplest things - definitions, basic concepts and simple operations. We assure you that the matrices will be understood by everyone who devotes at least a little time to them!
Matrix Definition
Matrix is a rectangular table of elements. Well, in simple terms – a table of numbers.
Typically, matrices are denoted in capital Latin letters. For example, matrix A , matrix B and so on. Matrices can be of different sizes: rectangular, square, and there are also row and column matrices called vectors. The size of the matrix is determined by the number of rows and columns. For example, let's write a rectangular matrix of size m on n , Where m – number of lines, and n – number of columns.
Items for which i=j (a11, a22, .. ) form the main diagonal of the matrix and are called diagonal.
What can you do with matrices? Add/Subtract, multiply by a number, multiply among themselves, transpose. Now about all these basic operations on matrices in order.
Matrix addition and subtraction operations
Let us immediately warn you that you can only add matrices of the same size. The result will be a matrix of the same size. Adding (or subtracting) matrices is simple - you just need to add up their corresponding elements . Let's give an example. Let's perform the addition of two matrices A and B of size two by two.
Subtraction is performed by analogy, only with the opposite sign.
Any matrix can be multiplied by an arbitrary number. To do this, you need to multiply each of its elements by this number. For example, let's multiply the matrix A from the first example by the number 5:
Matrix multiplication operation
Not all matrices can be multiplied together. For example, we have two matrices - A and B. They can be multiplied by each other only if the number of columns of matrix A is equal to the number of rows of matrix B. In this case each element of the resulting matrix, located in the i-th row and j-th column, will be equal to the sum of the products of the corresponding elements in the i-th row of the first factor and the j-th column of the second. To understand this algorithm, let's write down how two square matrices are multiplied:
And an example with real numbers. Let's multiply the matrices:
Matrix transpose operation
Matrix transposition is an operation where the corresponding rows and columns are swapped. For example, let's transpose the matrix A from the first example:
Matrix determinant
Determinant, or determinant, is one of the basic concepts of linear algebra. Once upon a time, people came up with linear equations, and after them they had to come up with a determinant. In the end, it’s up to you to deal with all this, so, the last push!
The determinant is a numerical characteristic of a square matrix, which is needed to solve many problems.
To calculate the determinant of the simplest square matrix, you need to calculate the difference between the products of the elements of the main and secondary diagonals.
The determinant of a matrix of first order, that is, consisting of one element, is equal to this element.
What if the matrix is three by three? This is more difficult, but you can cope.
For such a matrix, the value of the determinant is equal to the sum of the products of the elements of the main diagonal and the products of the elements lying on the triangles with a face parallel to the main diagonal, from which the product of the elements of the secondary diagonal and the product of the elements lying on the triangles with the face of the parallel secondary diagonal are subtracted.
Fortunately, in practice it is rarely necessary to calculate determinants of matrices of large sizes.
Here we looked at basic operations on matrices. Of course, in real life you may never encounter even a hint of a matrix system of equations, or, on the contrary, you may encounter much more complex cases when you really have to rack your brains. It is for such cases that professional student services exist. Ask for help, get a high-quality and detailed solution, enjoy academic success and free time.
