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✪ Vector norm. Part 4

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Definition

Let K be the main field (usually K = R or K = C ) and is the linear space of all matrices with m rows and n columns, consisting of elements of K . A norm is given on the space of matrices if each matrix is ​​associated with a non-negative real number ‖ A ‖ (\displaystyle \|A\|), called its norm, so that

In the case of square matrices (i.e. m = n), matrices can be multiplied without leaving the space, and therefore the norms in these spaces usually also satisfy the property submultiplicativity :

Submultiplicativity can also be performed for the norms of non-square matrices, but defined for several required sizes at once. Namely, if A is a matrix ℓ  ×  m, and B is the matrix m ×  n, That A B- matrix ℓ  ×  n .

Operator norms

An important class of matrix norms are operator norms, also referred to as subordinates or induced . The operator norm is uniquely constructed according to the two norms defined in and , based on the fact that any matrix m ×  n is represented by a linear operator from K n (\displaystyle K^(n)) V K m (\displaystyle K^(m)). Specifically,

Under the condition that norms on vector spaces are consistently specified, such a norm is submultiplicative (see ).

Examples of operator norms

Properties of the spectral norm:

1. The spectral norm of an operator is equal to the maximum singular value of this operator.
2. The spectral norm of a normal operator is equal to the absolute value of the maximum modulo eigenvalue of this operator.
3. The spectral norm does not change when a matrix is ​​multiplied by an orthogonal (unitary) matrix.

Non-operator norms of matrices

There are matrix norms that are not operator norms. The concept of non-operator norms of matrices was introduced by Yu. I. Lyubich and studied by G. R. Belitsky.

An example of a non-operator norm

For example, consider two different operator norms ‖ A ‖ 1 (\displaystyle \|A\|_(1)) And ‖ A ‖ 2 (\displaystyle \|A\|_(2)), such as row and column norms. Forming a new norm ‖ A ‖ = m a x (‖ A ‖ 1 , ‖ A ‖ 2) (\displaystyle \|A\|=max(\|A\|_(1),\|A\|_(2))). The new norm has an annular property ‖ A B ‖ ≤ ‖ A ‖ ‖ B ‖ (\displaystyle \|AB\|\leq \|A\|\|B\|), preserves the unit ‖ I ‖ = 1 (\displaystyle \|I\|=1) and is not operator .

Examples of norms

Vector p (\displaystyle p)-norm

Can be considered m × n (\displaystyle m\times n) matrix as a size vector m n (\displaystyle mn) and use standard vector norms:

Frobenius norm

Frobenius norm, or euclidean norm is a special case of the p-norm for p = 2 : ‖ A ‖ F = ∑ i = 1 m ∑ j = 1 n a i j 2 (\displaystyle \|A\|_(F)=(\sqrt (\sum _(i=1)^(m)\sum _(j =1)^(n)a_(ij)^(2)))).

The Frobenius norm is easy to calculate (compared to, for example, the spectral norm). It has the following properties:

• Submultiplicativity: ‖ A B ‖ F ≤ ‖ A ‖ F ‖ B ‖ F (\displaystyle \|AB\|_(F)\leq \|A\|_(F)\|B\|_(F)), because ‖ A B ‖ F 2 = ∑ i , j | ∑ k a i k b k j | 2 ≤ ∑ i , j (∑ k | a i k | | b k j |) 2 ≤ ∑ i , j (∑ k | a i k | 2 ∑ k | b k j | 2) = ∑ i , k | a i k | 2 ∑ k , j | b k j | 2 = ‖ A ‖ F 2 ‖ B ‖ F 2 (\displaystyle \|AB\|_(F)^(2)=\sum _(i,j)\left|\sum _(k)a_(ik) b_(kj)\right|^(2)\leq \sum _(i,j)\left(\sum _(k)|a_(ik)||b_(kj)|\right)^(2)\ leq \sum _(i,j)\left(\sum _(k)|a_(ik)|^(2)\sum _(k)|b_(kj)|^(2)\right)=\sum _(i,k)|a_(ik)|^(2)\sum _(k,j)|b_(kj)|^(2)=\|A\|_(F)^(2)\| B\|_(F)^(2)).
• ‖ A ‖ F 2 = t r ⁡ A ∗ A = t r ⁡ A A ∗ (\displaystyle \|A\|_(F)^(2)=\mathop (\rm (tr)) A^(*)A=\ mathop (\rm (tr)) AA^(*)), Where t r ⁡ A (\displaystyle \mathop (\rm (tr)) A)- matrix trace A (\displaystyle A), A ∗ (\displaystyle A^(*)) is a Hermitian conjugate matrix .
• ‖ A ‖ F 2 = ρ 1 2 + ρ 2 2 + ⋯ + ρ n 2 (\displaystyle \|A\|_(F)^(2)=\rho _(1)^(2)+\rho _ (2)^(2)+\dots +\rho _(n)^(2)), Where ρ 1 , ρ 2 , … , ρ n (\displaystyle \rho _(1),\rho _(2),\dots ,\rho _(n))- singular values ​​of the matrix A (\displaystyle A).
• ‖ A ‖ F (\displaystyle \|A\|_(F)) does not change when multiplying a matrix A (\displaystyle A) left or right onto orthogonal (unitary) matrices.

Module maximum

The modulo maximum norm is another special case of the p-norm for p = ∞ .

Norm Shatten

Consistency of matrix and vector norms

Matrix Norm ‖ ⋅ ‖ a b (\displaystyle \|\cdot \|_(ab)) on K m × n (\displaystyle K^(m\times n)) called agreed with the norms ‖ ⋅ ‖ a (\displaystyle \|\cdot \|_(a)) on K n (\displaystyle K^(n)) And ‖ ⋅ ‖ b (\displaystyle \|\cdot \|_(b)) on K m (\displaystyle K^(m)), If:

for any A ∈ K m × n , x ∈ K n (\displaystyle A\in K^(m\times n),x\in K^(n)). By construction, the operator norm is consistent with the original vector norm.

Examples of consistent but not subordinate matrix norms:

Equivalence of norms

All norms in space K m × n (\displaystyle K^(m\times n)) are equivalent, that is, for any two norms ‖ . α (\displaystyle \|.\|_(\alpha )) And ‖ . ‖ β (\displaystyle \|.\|_(\beta )) and for any matrix A ∈ K m × n (\displaystyle A\in K^(m\times n)) double inequality is true.

The eigenvalue problem is only defined for square matrices. In economic practice, it is often necessary to evaluate more than just square matrices. For such an estimate, one can use the universal concept of a norm, which is valid for matrices of any dimension.

Norma arbitrary matrix A is called a real number that satisfies a number of conditions, the most important of which are:

1.
, and
only in the case of a completely zero matrix A.

2.
, Where
.

Normal to some extent
can be figuratively represented as an indicator of the “thickness” or “power” of the matrix A.

The norm is called canonical , If

, i.e. it is not less than, modulo, any element of the matrix A. When choosing a norm, it is possible to use a variety of considerations that do not contradict the definition. However, in practice, the following canonical norms are usually sufficient:

1. m -norm
– summed, modulo, all lines matrices A

2. l -norm
– summed, modulo, all columns matrices A and the maximum of the amounts received is declared the norm.

3. k -norm
=
– the squares of all elements of the matrix are summed A and the root of this sum is declared to be the norm.

Vectors Basic definitions and concepts

A special case of a matrix consisting of one column has a wide independent application. The geometric representation of a vector by a directed segment, known from a school course, can be defined as a set of vector-segment projections written as a column matrix. Then we have the concept free vector , independent of the application point, which can be either at the origin ( radius vector ) and at any point in space. The direction of the vector is always strictly preserved. For the 2D case: = or = ; = or = . For generality, all projections are denoted below by X and are called coordinates vector. If any projection X is negative, then it is plotted in the opposite direction of the corresponding coordinate axis.

Vectors look exactly the same. = in 3D coordinate system - add coordinate z. But vectors of more than three dimensions are not visually representable - they can only be understood by analogy. General definition: vector V n-dimensional space is called an ordered set n coordinates = , the number of which is equal to the dimension of the space, i.e. n.

Vector length is determined by the formula d=
. All operations with vectors are the same as with matrices.

Consider linear combination three vectors: k +k +k .

If the equality k +k +k =0 is possible only for k=k =k =0, then the vectors ,And called linearly independent . Otherwise, at least one of the vectors can be expressed as the sum of the other two, and the vectors will be linearly dependent . For example, for k 0 can be written: =(- k -k ).

The maximum possible number of linearly independent vectors is dimensions space. So, for a plane, only two such vectors are possible, for a straight line - one. For n- dimensional space, the number of vectors is equal to n.

Let there be vectors on the plane , And . Let us show that they are linearly dependent. Let's make their linear combination: k + k + k = 0 and pass to the algebraic form:

.

Thus, putting k=1, we have: -+=0 or =+, i.e. the third vector is not independent and is expressed as the sum of the other two or decomposing along two other vectors. Consider the first two vectors in more detail: =A =A And =b=b. Then =With+d- very compact notation via unit vectors (or orts ). Let us show that the unit vectors are linearly independent: k+ k= k +k =0 or
, where k=k = 0.

Because With And d are arbitrary, then, obviously, any vector on the plane can be represented by a combination of two unit vectors and. This is called the expansion of a vector in terms of identity. basis or, more precisely, by orthonormal , because the length of each orth is equal to 1. Of course, one can expand not in terms of orts, but in terms of any two linearly independent vectors (in terms of the common basis ), For example, And , but the factorization in terms of orts is both simple and general.

All the concepts introduced above are valid for a space of any dimension. IN n-dimensional space always has n linearly independent vectors =,=,...,=, so any vector can be expanded in an orthonormal basis:= A 1 + A 2 +...+a n . The expansion of vectors in a basis of linearly independent vectors is always unique in any accepted basis.

» Lesson 12. Matrix rank. Matrix rank calculation. Matrix norm

Lesson number 12. Matrix rank. Matrix rank calculation. Matrix norm.

If all matrix minorsAorderkare equal to zero, then all minors of order k + 1, if such exist, are also equal to zero.
Matrix rank A is the largest order of the minors of the matrix A , other than zero.
The maximum rank can be equal to the minimum number of the number of rows or columns of the matrix, i.e. if the matrix has a size of 4x5, then the maximum rank will be 4.
The minimum rank of a matrix is ​​1, unless you are dealing with a zero matrix, where the rank is always zero.

The rank of a nondegenerate square matrix of order n is equal to n, since its determinant is a minor of order n and the nondegenerate matrix is ​​nonzero.
Transposing a matrix does not change its rank.

Let the rank of the matrix be . Then any minor of order , other than zero, is called basic minor.
Example. Given a matrix A.

The matrix determinant is zero.
Minor of the second order . Therefore, r(A)=2 and the minor is basic.
A basic minor is also a minor .
Minor , because =0, so it will not be basic.
Exercise: independently check which other second-order minors will be basic and which will not.

Finding the rank of a matrix by calculating all its minors requires too much computational work. (The reader can verify that there are 36 second-order minors in a fourth-order square matrix.) Therefore, a different algorithm is used to find the rank. To describe it, some additional information is required.

We call the following operations on them elementary transformations of matrices:
1) permutation of rows or columns;
2) multiplying a row or column by a non-zero number;
3) adding to one of the rows another row, multiplied by a number, or adding to one of the columns of another column, multiplied by a number.

Under elementary transformations, the rank of the matrix does not change.
Algorithm for calculating the rank of a matrix is similar to the determinant calculation algorithm and lies in the fact that with the help of elementary transformations the matrix is ​​reduced to a simple form for which it is not difficult to find the rank. Since the rank did not change with each transformation, by calculating the rank of the transformed matrix, we thereby find the rank of the original matrix.

Let it be required to calculate the rank of the matrix of dimensions mxn.

As a result of calculations, the matrix A1 has the form

If all rows, starting from the third one, are zero, then , since minor . Otherwise, by permuting rows and columns with numbers greater than two, we achieve that the third element of the third row is different from zero. Further, by adding the third row, multiplied by the corresponding numbers, to the rows with large numbers, we get zeros in the third column, starting from the fourth element, and so on.
At some stage, we will come to a matrix in which all rows, starting from (r + 1) th, are equal to zero (or absent at ), and the minor in the first rows and first columns is the determinant of a triangular matrix with non-zero elements on the diagonal . The rank of such a matrix is ​​. Therefore, Rang(A)=r.

In the proposed algorithm for finding the rank of a matrix, all calculations must be performed without rounding. An arbitrarily small change in at least one of the elements of the intermediate matrices can lead to the fact that the resulting answer will differ from the rank of the original matrix by several units.
If the elements in the original matrix were integers, then it is convenient to perform calculations without using fractions. Therefore, at each stage, it is advisable to multiply the strings by such numbers that no fractions appear in the calculations.

In laboratory and practical work, we will consider an example of finding the rank of a matrix.

FINDING ALGORITHM MATRIX REGULATIONS .
There are only three norms of the matrix.
First Matrix Norm= the maximum of the numbers obtained by adding all the elements of each column, taken modulo.
Example: let a 3x2 matrix A be given (Fig. 10). The first column contains elements: 8, 3, 8. All elements are positive. Let's find their sum: 8+3+8=19. The second column contains the elements: 8, -2, -8. Two elements are negative, therefore, when adding these numbers, it is necessary to substitute the modulus of these numbers (that is, without the minus signs). Let's find their sum: 8+2+8=18. The maximum of these two numbers is 19. So the first norm of the matrix is ​​\u200b\u200b19.

Figure 10.

Second Matrix Norm is the square root of the sum of the squares of all matrix elements. And this means we square all the elements of the matrix, then add the resulting values ​​\u200b\u200band extract the square root from the result.
In our case, the 2 norm of the matrix turned out to be equal to the square root of 269. In the diagram, I roughly took the square root of 269 and the result was approximately 16.401. Although it is more correct not to extract the root.

Third Norm Matrix is the maximum of the numbers obtained by adding all the elements of each row, taken modulo.
In our example: the first line contains elements: 8, 8. All elements are positive. Let's find their sum: 8+8=16. The second line contains elements: 3, -2. One of the elements is negative, so when adding these numbers, you must substitute the modulus of this number. Let's find their sum: 3+2=5. The third line contains the elements 8, and -8. One of the elements is negative, so when adding these numbers, you must substitute the modulus of this number. Let's find their sum: 8+8=16. The maximum of these three numbers is 16. So the third norm of the matrix is ​​\u200b\u200b16.

Compiled by: Saliy N.A.