Matrix Eigenvalue range, eigenvalue of the matrix

This is a typical operation in a matrix:

Step 1: Calculate the determinant |into e-a|Into -5

Into +1 Into -9 2

+7|, which is where the most computational effort lies.

Step 2: Factor to find the eigenvalues (usually real numbers) Step 3: Substitute the eigenvalues one by one.

into e-ax=0

Find the basic solution. Linearally independent eigenvectors can be obtained.

Calculations with MATLAB software are fine.

Run after installation.

Enter a=[ ; Enter.

Then enter eig(a) and press enter to get the eigenvalue of .

Calculate the absolute value of e-a, and calculate this determinant to solve three solutions, which are three eigenvalues.

a=[ ;

Three eigenvalues are obtained directly:

a=[ ;a =

eig(a) to obtain three eigenvalues. ans =

Let the friend ruler a be a square matrix of order n, and if there is a number m and a non-zero n-dimensional column vector x such that ax=mx holds, then m is said to be an eigenvalue of matrix a.

The non-zero vector x is called the eigenvector corresponding to the eigenvalue of a.

In mathematics, a matrix is a set of complex or real numbers arranged in a rectangular array, which originally came from the square matrix formed by the coefficients and constants of the system of equations. This concept was first proposed by the 19th-century British mathematician Kelly.

Matrices are a common tool in advanced algebra and are also commonly found in applied mathematics disciplines such as statistical analysis. In physics, matrices are used in circuits, mechanics, optics, and quantum physics.

have applications in ; Computer science.

, 3D animation.

Crafting also requires the use of matrices. The operation of the matrix is numerical analysis.

important questions in the field. Decomposing matrices into combinations of simple matrices can simplify the operation of matrices in theory and practical applications. For some matrices with wide applications and special forms, such as sparse matrices and quasi-diagonal matrices, there are specific fast operation algorithms.

Property 1: All eigenroots of the n-order phalanx a=(aij) are 1, 2,...,n (including heavy roots, then:

Property 2: If is a eigenroot of the invertible matrix a, and x is the corresponding eigenvector, then 1 is a eigenroot of the inverse of a, and x is still the corresponding eigenvector.

Property 3: If is a eigenroot of the matrix a, and x is the corresponding eigenvector, then m to the power is a eigenroot of a, and x is still the corresponding eigenvector.

Nature 4: Set 1, 2 ,..., m is the eigenvalue of the different beams of the square matrix a. xj is a feature vector belonging to i ( i=1,2,..., m), then x1, x2 ,..., xm is linearly independent, i.e., eigenvectors that are not the same eigenvalues are linearly independent.

If the multiple of the eigenvalue a is k, then n-r(a) sets a as an nth-order matrix, and according to the relation ax x, (e a) x 0 can be written, and then the eigenpolynomial e a 0 can be written, and the matrix a has n eigenvalues (including the heavy eigenvalues). Substituting the eigenvalue i into the original eigenpolynomial solves the equation (i.e.) x 0, and the solution vector x is the eigenvector of the corresponding eigenvalue i.

Notes:

Generalized eigenvalues: If the eigenvalues are generalized to the plural collar domain, the generalized eigenvalues are in the form of: a b

where A and B are pure formations. By solving the equation (a-b) = 0 to obtain the generalized eigenvalue silver and , the determinant (a-b) = 0 (where the determinant is the determinant) forms a matrix set, such as a-b. The plural noun in the eigenvalue is called "pencil".

If b is reversible, then the original relation can be written as a standard eigenvalue problem. When b is an irreversible matrix (which cannot be inversely transformed), the generalized eigenvalue problem should be solved in its original form.

Let a be an n-order square, and if there is a number m and a non-zero n-dimensional column vector x such that ax=mx holds, then m is said to be an eigenvalue of the matrix a orEigenvalues

The formula ax = x can also be written as ( a- e) x = 0. This is a homogeneous linear equation system of n unknowns, n equations, and hu of the spine.

It has sufficient and necessary conditions for a non-zero solution.

is the coefficient determinant | a-λe|=0。

Matrix eigenvalues.

Property 1: If is a eigenroot of the reversible matrix a, x is the corresponding eigenvector.

Then 1 is an eigenroot of the inverse of a, and x is still the corresponding eigenvector.

Property 2: If is a eigenroot of the matrix a, and x is the corresponding eigenvector, then the m power is a eigenroot of a's m power, and x is still the corresponding eigenvector.

Nature 3: Set 1, 2 ,..., m is the distinct eigenvalue of the square matrix a. xj is a feature vector belonging to i ( i=1,2,..., m), then x1, x2 ,..., xm linear Biplex independence, i.e., eigenvectors that are not the same eigenvalues are linearly independent.

The matrix eigenvalues are as follows:

Let a be an n-order square, and if there are several m and a non-zero n-dimensional column vector x such that ax=mx holds, then m is said to be an eigenvalue or eigenvalue of the matrix a.

The formula ax = x can also be written as ( a- e) x = 0. This is a system of homogeneous linear equations for n unknowns n square lead finger ranges, and a sufficient and necessary condition for it to have a non-zero solution is the coefficient determinant | a-λe|=0。

Properties of matrix eigenvalues:

Property 1: If Huai is a eigenroot of the invertible matrix a, and x is the corresponding eigenvector, then 1 is a eigenroot of the inverse of a, and x is still the corresponding eigenvector.

Property 2: If is an eigenroot of the chain filial piety matrix a, and x is the corresponding eigenvector, then m to the power is an eigenroot of a, and x is still the corresponding eigenvector.

Nature 3: Set 1, 2 ,..., m is the distinct eigenvalue of the square matrix a. xj is a feature vector belonging to i ( i=1,2,..., m), then x1, x2 ,..., xm is linearly independent, i.e., eigenvectors that are not the same eigenvalues are linearly independent.

Eigenvectors are vectors corresponding to the eigenvalues of matrices, and they are also an important concept in linear algebra. In mathematics, the eigenvectors and eigenvalues of matrices form the spectrum of matrices, which are the basis of matrix feature decomposition. Feature vectors are also widely used in machine learning and deep learning.

1.The premise of solving eigenvectors is to find the eigenvalues first. Let the matrix a be an nth-order square matrix, then the eigenvalues satisfy the following eigenequation:

a - i |0, where i is the identity matrix and | a - i |is the determinant of the matrix a - i, and solving this equation gives all the eigenvalues of matrix a 1, 2, 、..Carry away n.

2.For each eigenvalue i, there is a corresponding eigenvector UI, i.e., aui = iui. Therefore, the method of finding eigenvectors can be transformed into solving the problem of solving the system of linear equations aui = iui.

3.For AUI = IUI, since UI ≠ 0, the equation can be translated into (A - II) UI = 0, and (A - II) as a matrix of coefficients gives a homogeneous linear equation. Therefore, the homogeneous linear equations can be solved by Gaussian elimination method or LU decomposition, so as to solve the eigenvectors.

4.Due to the presence of non-zero vectors in the null space of the matrix, there may be multiple linearly independent eigenvectors for some scattered eigenvalues. When calculating the eigenvectors, it is necessary to pay attention to the selection of linearly independent vectors, and the system of sublinear equations of the Qichong hidden rows can be simplified by the basic row transformation or the Gaussian Jordan elimination method to obtain the linear independent vectors, so as to solve the eigenvectors.

In short, the eigenvector needs to be solved first, and then the eigenvalue is substituted into the linear equation system to solve the eigenvector, and attention is paid to the selection of linearly independent vectors. Feature vector solving has a wide range of applications in areas such as machine learning to help us better understand and process data.

Let a be an n-order square, and if there is a number m and a non-zero n-dimensional column vector x such that ax=mx holds, then m is said to be a characteristic value or eigenvalue of a. The non-zero n-dimensional column vector x is called the eigenvector or eigenvector of the matrix a that belongs to (corresponds to) the eigenvalue m, or is referred to as the eigenvector of a or the eigenvector of a.

The eigenvalue of the matrix is: let a be a square matrix of order n, and if there is a number m and a non-zero n-dimensional column vector x such that ax=mx holds, then m is said to be an eigenvalue or eigenvalue of matrix a.

Let a be a square matrix of order n, and if the number and the n-dimensional non-zero column vector x make the relation ax = x hold, then such a number is called the eigenvalue of matrix a, and the non-zero vector x is called the eigenvector corresponding to the eigenvalue of a. The formula ax = x can also be written as (A: a- e) x=0. This is a system of homogeneous linear equations of n unknowns n equations, and the sufficient and necessary condition for it to have a non-zero solution is the coefficient determinant | a-λe|=0。

Let a be an nth-order matrix on the number field p, which is an unknown quantity with a smile, and the coefficient determinant |a-λe|A characteristic polynomial called a is denoted ( e-a|, is a polynomial of the nth order on p with respect to , and e is the identity matrix.

(λe-a|=λn+a1λn-1+…+an= 0 is an n-order algebraic equation called the eigenequation of a. Characteristic equation (e-a|The root of =0 (e.g., 0) is called the eigenroot (or eigenvalue) of a.

N-order algebraic equations have and only n roots in the complex number domain, but not necessarily in the real number domain, so the number and absence of eigenroots are not only related to a, but also to the number field p.

Substituting the eigenvalue of a 0 into (e-a)x=0 yields a system of equations (0e-a)x=0, which is a homogeneous system of equations, called a eigenequation system of about 0. Because |λ0e-a|=0, ( 0e-a)x=0 There must be a non-zero solution, which is called the eigenvector of a belonging to 0. The eigenvectors of all 0's make up the eigenvector space of 0 as a whole.