Matrix (mathematics)

A matrix is ​​a mathematical object written as a rectangular table of ring or field elements, integer or complex numbers, which is a collection of rows and columns at the intersection of which its elements are located. The number of rows and columns of the matrix determines the size of the matrix. Although, for example, triangular matrices have historically been considered, nowadays one speaks exclusively of rectangular matrices, since they are the most convenient and general.

Matrices are widely used in mathematics for the compact representation of systems of linear algebraic or differential equations. In this case, the number of matrix rows corresponds to the number of equations, and the number of columns corresponds to the number of unknowns. As a result, the solution of systems of linear equations is reduced to operations on matrices.

Matrices allow the following algebraic operations:

-addition of matrices having the same size;

-multiplication of matrices of suitable size (a matrix with n columns can be multiplied on the right by a matrix with n rows);

-multiplying a matrix by an element of the main ring or field.

With respect to addition, matrices form an abelian group. If we also consider multiplication by a scalar, then the matrices form a module over the corresponding ring (vector space over the field). The set of square matrices is closed under matrix multiplication, so square matrices of the same size form an associative ring with unity under matrix addition and matrix multiplication.

The matrix is ​​the matrix of some linear operator. The properties of the matrix correspond to the properties of the linear operator. In particular, the eigenvalues ​​of a matrix are the eigenvalues ​​of the operator corresponding to the corresponding eigenvectors.

In mathematics, many different types and types of matrices are considered. For example, identity, symmetric, skew-symmetric, upper triangular (lower triangular) matrices.

Of particular importance in matrix theory are various normal forms, that is, the canonical form, to which a matrix can be reduced by changing coordinates. The most important (in the theoretical sense) and elaborated is the theory of Jordan normal forms. In practice, however, normal forms are used that have additional properties, such as stability.

For the first time, matrices were mentioned in ancient China, then called the “magic square”. The main application of matrices was the solution of linear equations. Magic squares were known a little later among Arab mathematicians, around then the principle of matrix addition appeared. After developing the theory of determinants in the late 17th century, Gabriel Cramer began developing his theory in the 18th century and published Cramer's rule in 1751. Approximately in the same period of time, the “Gauss method” appeared. Matrix theory began its existence in the middle of the 19th century in the works of William Hamilton and Arthur Cayley. Fundamental results in matrix theory are due to Weierstrass, Jordan, Frobenius. The term "matrix" was coined by James Sylvester in 1850.

Let M = {1,2,…,m} and N = {1,2,...,n}, where m, n ∈ N, are two finite sets.

We call an m × n matrix (read m by n) with elements from some ring or field K a mapping of the form

A : M × N → K.

If index i runs through the set M, and j runs through the set N, then the element A (i, j) turns out to be a matrix element located at the intersection of the i-th row and the j-th column:

- The i-th row of the matrix consists of elements of the form A (i, j), where j runs through the entire set N;

- the j-th column of the matrix consists of elements of the form A (i, j), where i runs through the entire set M.

So an m × n matrix consists exactly of

- m lines (n elements each)

- and n columns (m elements each).

According to this

- each row of the matrix can be interpreted as a vector in the n-dimensional coordinate space Kⁿ;

- each column of the matrix - as a vector in the m-dimensional coordinate space K^m.

The matrix itself is naturally interpreted as a vector in the space K mn with dimension mn. This allows one to introduce component-by-component addition of matrices and multiplication of a matrix by a number, as far as matrix multiplication is concerned, it essentially relies on the rectangular structure of the matrix.

If a matrix has the same number of rows m as the number of columns n, then such a matrix is ​​called a square matrix, and the number m = n is called the size of a square matrix or its order.

A matrix is ​​usually denoted by a capital letter of the Latin alphabet: let

A : M × N → K.

then A is a matrix, which is interpreted as a rectangular array of elements of the field K of the form a_ij = A (I, j), where

- the first index means the row index: I = 1,m;

- the second index means the column index: j= 1, n;

thus, a_ij is an element of the matrix A located at the intersection of the i-th row and the j-th column. In accordance with this, the following compact notation for an m × n matrix is ​​adopted:

A=(a_ij)^m,n_{i=1, j=1},

or simply:

A = (a_ij),

if you just need to specify the designation for the elements of the matrix.

Sometimes, instead of a_ij, they write a_i,j to separate the indices from each other and avoid confusion with the product of two numbers.

Associated with each m × n matrix A = (a_ij) is an n × m matrix B = (b_ij) of the form

B_ij= a_ji, I=1, m, j=1, n.

Such a matrix is called the transposed matrix for A and is denoted as A^T. A transposed matrix can be obtained by swapping the rows and columns of the matrix. The m × n matrix A = (a_ij) becomes an n × m matrix under this transformation.

Matrices of size m × 1 and 1 × n are elements of the spaces K^m and Kⁿ, respectively:

an m × 1 matrix is called a column vector and has a special notation:

colon ( a₁ ,…, a_i ,…,a_m ) = (a₁,…,a_i,…,a_m)^T;

matrix of size 1 × n is called a row vector and has a special notation:

row (a₁,…,a_i,…,a_n) = (a₁,…,a_i,…,a_n);

Multiplying a Matrix by a Number

Multiplication of a matrix A by a number λ (notation: λA) consists in constructing a matrix B, the elements of which are obtained by multiplying each element of the matrix A by this number, that is, each element of the matrix B is equal to

B_ij = λa_ij

Matrix addition

The addition of matrices A + B is the operation of finding a matrix C, all of whose elements are equal to the pairwise sum of all the corresponding elements of matrices A and B, that is, each element of matrix C is equal to

C_ij = a_ij + b_ij

Matrix multiplication (notation: AB, rarely with the multiplication sign A × B) is the operation of calculating the matrix C, the elements of which are equal to the sum of the products of the elements in the corresponding row of the first factor and the column of the second.

C_ij = ∑ⁿ_k=1 a_ik b_kj

The number of columns in matrix A must match the number of rows in matrix B. If matrix A has dimensions m × n, B - n × k, then the dimension of their product AB = C is m × k.

If the elements of the matrix A = (a_ij) are complex numbers, then the complex conjugate (not to be confused with the Hermitian conjugate! matrix is equal to A = (a_i,j). Here a is the number complex conjugate to a.

Transposition has already been discussed above: if A = (a_ij), then A^T = (a_ji). For complex matrices, the Hermitian conjugation is more common: A* = A^-T. From the point of view of the operator view of matrices, the transposed and Hermitian conjugate matrix are the matrices of the operator conjugate with respect to the scalar or Hermitian product, respectively.

Linear Combinations

In a vector space, a linear combination of vectors Х₁,…..,Х_n is a vector

X = a₁x₁ +......+ a_nx_n,

where a₁ …..,a_n are the expansion coefficients:

- if all coefficients are equal to zero, then such a combination is called trivial,

- if at least one coefficient is different from zero, then such a combination is called non-trivial.

This allows us to describe the product C = AB of matrices A and B in terms of linear combinations:

- columns of matrix C are linear combinations of columns of matrix A with coefficients taken from matrix B;

- rows of matrix C are linear combinations of rows of matrix B with coefficients taken from matrix A.

If any vector can be represented as a linear combination, then one speaks of a linear dependence of this vector on the elements of the combination.

Some set of elements of a vector space is called linearly dependent if there exists a linear combination of elements of this set equal to zero, or

0 =a₁x₁ +…….+a_nx_n,

where not all numbers a₁ ….,a_n are equal to zero if such a non-trivial combination does not exist, then the given set of vectors is called linearly independent.

The linear dependence of the vectors means that some vector of a given set is linearly expressed through the rest of the vectors.

Each matrix is ​​a collection of vectors (of the same space). Two such matrices are two sets. If each vector of one set is linearly expressed in terms of the vectors of another set, then in the language of matrix theory this fact is described using the product of matrices:

- if the rows of the matrix C are linearly dependent on the rows of the matrix B, then C = AB for some matrix A;

- if the columns of matrix C are linearly dependent on the columns of another matrix A, then C = AB for some matrix B.

The number of linearly independent rows of the matrix is called the row rank of the matrix, and the number of linearly independent columns of the matrix is called the column rank of the matrix. In fact, both ranks are the same. Their common value is called the rank of the matrix.

The rank of a matrix is equal to the number of non-zero rows.

Another approach that is equivalent to this is to define the rank of a matrix as the maximum order of a non-zero matrix minor.

Matrix operations. Addition and subtraction are allowed only for matrices of the same size.

There is a zero matrix Θ such that its addition to another matrix A does not change A, i.e.

A + Θ = A

All elements of the zero matrix are equal to zero.

Only square matrices can be raised to a power.

- Associativity of addition: A + (B + C) = (A + B) + C.

- Commutativity of addition: A + B = B + A.

- Associativity of multiplication: A(BC) = (AB)C.

- Matrix multiplication is non-commutative: AB ≠ BA. Using this property, a matrix commutator is introduced.

Distributivity of multiplication with respect to addition:

A(B + C) = AB + AC;

(B + C)A = BA + CA.

Given the properties mentioned above, matrices form a ring with respect to addition and multiplication operations.

Properties of the matrix transposition operation:

(A^T)^T = A

(AB)^T = B^T A^T

(A^-1)T = (A^T)^-1 if the inverse matrix A^-1 exists.

(A + B)^T = A^T + B^T

detA = detA^T

The following transformations are called elementary transformations of matrix rows:

- Multiplying a string by a non-zero number,

- Adding one string multiplied by a number to another string,

- Rearrangement of two lines.

Elementary matrix column transformations are defined similarly. Under elementary transformations, the rank of the matrix does not change.

The matrix of a linear operator is a matrix that expresses a linear operator in some basis. In order to obtain it, it is necessary to act with the operator on the basis vectors and write the coordinates of the obtained vectors (images of basis vectors) into the columns of the matrix.

The matrix of an operator is similar to the coordinates of a vector. In this case, the action of the operator on a vector is equivalent to multiplying the matrix by the column of coordinates of this vector in the same basis.

We choose a basis e_k. Let x be an arbitrary vector. Then it can be expanded in this basis:

x = x^ke_k,

where x^k are the coordinates of the vector x in the chosen basis.

Here and below, summation over silent indices is assumed.

Let A be an arbitrary linear operator. We act on both sides of the previous equality, we get

Ax = x^k Ae_k.

We also expand the vectors Ae_k in the chosen basis, we obtain

Ae_k = a^j_k e_j,

where a^j_k is the j-th coordinate of the k-th vector from Ae_k.

Substituting the expansion into the previous formula, we get

Ax = x^ka^j_k e_j = (a^j_k x^k)e_j.

The expression a^j_k x^k, enclosed in brackets, is nothing else than the formula for multiplying a matrix by a column, and thus the matrix a^j_k, when multiplied by a column x^k, results in the coordinates of the vector A_x resulting from the action of the operator A on the vector x, which is needed to get.

If we swap a pair of columns or rows in the resulting matrix, then we will get another matrix corresponding to the same set of basic elements e_k. In other words, the order of the basic elements is assumed to be strictly ordered.

Matrices play an important role in group theory. They are used in the construction of general linear groups, special linear groups, diagonal groups, triangular groups, unitriangular groups.