Matrix analysis (1) norm in fourth chapter

This chapter is mainly about matrix functions.

1. Norm

One.

The absolute value of what is usually said is a norm.

The norm is the extension of the concept of absolute value, which is designed to measure a vector in a certain way. such as the length of two-dimensional coordinates.

Definition of Norm:

Set V is a linear space on a number of fields F, if any x∈v, corresponds to a number | | x| | Meet:

Positive characterization: | | x| | ≥0, and | | x| | =0 when and only as x=0; Homogeneous: Any k∈f, x∈v, | | kx| | = |k| | | x| |; Trigonometric inequality: Any x,y∈v, with | | x+y| | ≤| | x| | +|| y| |.

It is said that V is the normed linear space, | | x| | The norm called X. These are also the three axioms of the norm. The general discussion is finite dimensional space.

Common norms are: 1 norm (sum of the absolute values of each element of the vector), Infinity norm (absolute value of the element with the highest absolute value in the vector element), 2 norm (the sum of the squares of the absolute value of all elements is 1/2 square, that is, the unitary space is induced by the inner product norm.)

The P-norm is: The square in the 2 norm is replaced by the P-order, and the last 1/2 is replaced by the 1/p, which is the P-norm. By definition, they can be proved to meet norm conditions.

Basic inequalities in two functional analysis:

Holder inequalities: Set p,q>1, and 1/p + 1/q = 1, then any x,y∈v, there is: Σ (i=1,n) |xi * yi| P-Norm of the P-norm * y of the ≤x.

Minkowski inequalities: P-norm of P-norm + y for p≥1, p-norm ≤x of X+y.

Equivalence: Any two norm A, B, if there are positive K1 and K2, so that any x∈v have k1| | x| | (A-norm) ≤| | x| | (b-norm) ≤k2| | x| | (A-norm), A and B are called equivalent.

Any two norms in a finite dimensional linear space are equivalent.

Convergence: Set x1,x2,... xm, ... Is the sequence of elements in V, which, if x∈v exists, makes Lim (m→∞) | | Xm-x | | (a norm) = 0, the sequence {XM} converges to X by a a-norm.

Because of any two norm equivalents in the finite dimension V, if {xm} converges to x0 by a certain norm, it converges to x0 by any norm.

Two.

Norm of the Matrix:

To migrate the norm of vectors to a matrix, there are the following definitions:

If any m x N-order matrices are a∈c, there are real numbers corresponding to them, remember | | a| |, and satisfies: positive definite, homogeneous, triangular unequal, is called | | a| | is the vector norm or generalized matrix norm of matrix A.

Given the multiplication of the matrix, add a condition to be defined:

If the a∈c of any N-n-order Phalanx , there are real numbers corresponding to them, remember | | a| |, and satisfies: positive definite, homogeneous, triangular, compatibility (any A, B, | | ab| | ≤| | a| | || b| |), the name | | a| | is the matrix norm (or product norm) of matrix A.

Frobenius norm (f-norm) of matrices:

Square A, then | | a| | (f-norm) = (σ|aij|²) ^1/2 = (tr (a^h A)) ^1/2.

The F-norm has many good properties.

1) Set A is n x N square, x is n-dimensional vector, then | | ax| | (2 norm) ≤| | a| | (f-norm) | | x| | (2 norm);

2) Any unitary array u,v, The matrix A is: | | ua| | (f-norm) = | | av| | (f-norm) = | | a| | (f-norm).

Three.

The vector norm is compatible with the matrix norm: (This omits the norm in parentheses by two words.)

Compatibility: If the matrix A and vector x, vector norm | | x| | (v) and Matrix norm | | a| | (m) satisfying inequalities: | | ax| | (v) ≤| | a| | (M) | | x| | (v), the matrix norm and the vector norm are compatible.

Any two kinds of matrix norm are equivalent, and any two vector norm is also equivalent. So is there a vector norm that is compatible with any given matrix norm? What's the reverse?

1) Set | | a| | is a matrix norm on C, there must be a vector norm that is compatible with it.

2) Set | | x| | (v) is a vector norm on C, then to any A, define | | a| | = Max (| | x| | (v) = 1) | | ax| | (v), then | | a| | is A with | | x| | (v) a compatible matrix norm. Call this matrix norm to be subordinate to the vector norm | | x| | operator Norm of (v).

For any A and x, from the three norm of the vector x: 1 norm, 2 norm, the operator norm of the Infinity norm is: | | a| | (1) Column norm (maximum value of the sum of the absolute values of each column element), | | a| | (2) Spectral norm (1/2 of the maximum eigenvalue of a^h a), | | a| | (∞) Line norm (the maximum value of the sum of the absolute values of the elements per row).

- || a| | (F) and | | x| | (2) is compatible, | | a| | (2) with | | x| | (2) Of course also compatible, but | | a| | (2) ≤| | a| | (F).

-if there is a constant m, so that any x has | | ax| | (A-norm) ≤m| | x| | (a norm), then | | a| | (a norm) ≤m. In other words, Vector norm | | x| | (a) The corresponding operator norm | | a| | (a) is the smallest m that causes the first inequality to be established.

-Generally, the calculation of operator norm is very complicated, there is no general calculation method.

Four.

Some theorems and applications:

1) square A, | | a| | is the matrix norm, if | | a| | <1, then i-a non-singular, and | | (i-a) ^-1 | | ≤| | i| | /(1-| | a| |) .

2) The largest of the n eigenvalues of A is known as the spectral radius of a. The spectral radius of a is not greater than any of the matrix norms of a.

3) Any real number e>0, there is a certain norm of a < a spectral radius +e.