Matrix analysis of the fourth chapter (2) Eigenvalue estimation, matrix progression

I. Eigenvalue estimation

Eigenvalue is a very important property of the matrix, when the order is too high, it is difficult to calculate the eigenvalue, so it needs to be estimated.

The contents of the norm are described in matrix Analysis (1).

Theorem 1: Set A's eigenvalues to λ1,λ2,.. Λn. Then |λi| ≤| | a| |, where the matrix norm is the row norm and the column norm. and |λi|²≤| | a| |, where the matrix norm is the spectral norm.

Define the Gaelic disk (Gerschgorin): Square a = (AIJ), so that δil = the sum of the absolute values of the line I elements in a-|aii|. That is, Δil is the sum of the absolute values of the elements except the diagonal element for line I. Guerre gi is a disc with AII as the center and a radius of δil.

A There are n Gaelic circles.

Theorem 2:a n Guerre G1, G2,.. Gn, with the following characteristics:

1) Any characteristic value of a λ∈∪ (I=1, N) Gi.

2) within the isolated Gaelic circle there is only one characteristic value, within the Gaelic circle of Unicom, several Gaelic circular unicom have several eigenvalues.

The features of the Gaelic circle can be summed up as follows:

1. If the origin is not within the Gaelic circle of a, then a is not singular.

2. If a diagonally dominant, that is |aii| >δi, (including row diagonally dominant and column diagonally dominant), then a is not singular.

3. If the n Gaelic Circle 22 of a does not intersect, A has n mutually exclusive eigenvalues, and A is a simple matrix.

4. If the solid phalanx A has k isolated Gaelic circles, then A has at least k distinct real eigenvalues. In fact, the center of the N Gaelic circle of A is on the real axis, each isolated Guerre has only one eigenvalue, and if the real Phalanx has complex eigenvalues, it must appear in pairs.

Because the isolated Guerre has good properties, you can use the following method to indent the Guerre:

1) Use Guerre theorem for a^t. (generally no use)

2) Select the appropriate positive d1,d2,... dn, make D = diag{d1, d2, ... dn}, B = d a d^-1, then the Gale Center of B is unchanged, A and B are similar, with the same eigenvalues, Di is selected by:

If the di<1, the remaining is 1, then a of the first Gaelic circle narrowed, the rest enlarged;

If di>1, the remainder is 1, then A's the I-Gaelic circle enlarges, the remainder shrinks.

Two. Matrix progression:

In mathematical analysis, the series is discussed based on the sequence limit theory. The series in The matrix is similar.

The definition of convergence of matrix A in A0 is that each element in a converges to the corresponding element in A0.

Definition 1: set {Ak}, m x n order matrix Ak∈c, called σ (k=1,∞) Ak for matrix progression. Make SN =σ (K=1,N) AK, if {Sk} converges and has the limit s, then the series Σ (k=1,∞) AK converges and S.

-The concept of absolute convergence is similar, and the absolute convergence of the elements of each element is absolutely convergent.

A power series of matrices: a series of matrices such as σ (m=0,∞) cm a^m, called a. (M is subscript and number of times).

Theorem: Set the complex variable power series Σ (m=0,∞) cm z^m The convergence radius of R, the spectral radius of square A is ρ (a), then:

When ρ (A) < R, the matrix power series is absolutely convergent;

When ρ (A) > R, the Matrix power series method is vector.

Inference: Σ (m=0,∞) a^m convergence is equivalent to ρ (A) < 1, at this time its and is (I-A) ^-1.