The master used MATLAB to help me calculate the maximum eigenvalue of the matrix

The elements on the diagonal of d calculated below are eigenvalues, up to .

Each column of v is a feature vector and is normalized (the last sentence is to test the normalization).

a=[1 5 1/3; 1/5 1 1/2; 3 2 1];

v d]=eig(a)

v = -

d =0 + 0

sum(abs(v).^2)ans =

There are n eigenvalues of n-order matrices, and the largest value is the maximum eigenvalue.

Step 1: Compute the feature polynomial;

Step 2: Find all the roots of the eigenequation of the slow hand, that is, all the eigenvalues;

Step 3: For each eigenvalue, find the system of homogeneous linear equations: a basic solution system, then you can find all the eigenvectors that belong to the eigenvalues. Find the slip of the eigenvector:

Let a be an nth-order matrix, and according to the relation ax = x, ( e-a)x=0 can be written, and then the eigenpolynomial | can be writtenλe-a|=0, you can find that matrix a has n eigenvalues (which one includes heavy eigenvalues). Substituting the eigenvalue i into the original eigenpolynomial, the equation (ie-a) x=0 is solved, and the solution vector x is the eigenvector of the corresponding eigenvalue i.

Sufficient and necessary conditions for judging that the matrix can be diagonalized:

There are two sufficient and necessary conditions for matrices to be diagonalized:

1. The matrix has n different eigenvectors;

2. The multiplicity of the heavy root of the eigenvector is equal to the number of the basic solution system. For the second sufficient condition, there needs to be more than two duplicate eigenvalues verifiable (one is equivalent to no double root).

If the matrix a can be diagonalized, then the main diagonal elements of the diagonal matrix are all eigenvalues of a, and the rest of the elements are all 0. (The diagonal array of a matrix is not unique, and its eigenvalues can be reversed, but there is an invertible matrix p composed of the corresponding eigenvector order so that pap= )

Similarity matrices have the same eigenvalues. For both a and b =2, the remaining quadratic terms are solved according to the pending coefficient method.

The algorithm for the maximum eigenvalue of the matrix is calculated according to the equation ax = x.

The maximum eigenvalue of a matrix is the largest number of all eigenvalues in the matrix. To find it, you need to find all the eigenvalues of the matrix and then compare their sizes. All eigenvalues of the matrix refer to the number satisfying the equation ax = x, where a is an n-order square matrix and x is a non-zero n-dimensional column vector.

The specific steps to find them are: first, find the eigenpolynomial of matrix a, i.e., the determinant|λe-a|where e is the identity matrix. Then find the root of the eigenpolynomial, i.e., the equation|λe-a|=0, these roots are the eigenvalues of matrix a.

Finally, for each eigenvalue, find the non-zero solutions of the square split range group (e-a)x=0, which are the eigenvectors corresponding to the eigenvalues.

In this way, all the eigenvalues of matrix a and the corresponding eigenvectors can be obtained. Then, compare the magnitudes of all the eigenvalues and find the largest one, which is the maximum eigenvalue of matrix a.

Applications:

1. In algebra, the eigenvalues of the matrix together with the corresponding eigenvectors constitute a description of the essential properties of the matrix. For example, the sign of the eigenvalue determines the sign type of the matrix, while the eigenvector can provide key information.

2. In differential equations, eigenvalues are usually defined as the roots that make the solution of the corresponding homogeneous linear differential equation satisfy certain boundary conditions. By solving for these eigenvalues, we can obtain the physical meaning or properties related to the geometry of a particular region, material properties, etc.

3. In the field of numerical computing and machine learning, eigenvalues are widely used. For example, we can use eigenvalues to reduce the dimensionality of data in order to process and analyze large amounts of high-dimensional data more efficiently.

4. In the field of physics and engineering, eigenvalues can be used for problems such as vibration analysis, determination of the orientation of the principal number of inertia matrices, and stress tensor analysis. By solving the resulting eigenvalues, we can derive physical information about the system.

We often use MATLAB to perform matrix operations, so how do we find the eigenvalues of matrices? Let me share it with you.

matlab

01 First of all, we open the MATLAB software and define a matrix, as shown in the figure below.

02 Then execute the [x,y] = eig(a) formula to find the value of x,y, as shown in the figure below.

03 Next, diag(y) is executed to find the eigenvalue of y, as shown in the following figure.

04 The eigenvalues of the last matrix are marked in the red box, as shown in the figure below.

The traditional way to find the eigenvalue is to make the eigenpolynomial ae a 0 to find the eigenvalue of a, for any eigenvalue h of a, all non-zero solutions of the eigenequation (ae a) x 0 x are the eigenvectors of the eigenvalue n of the matrix a, and the calculation of the two is to divide the bench, one is to calculate the determinant, and the other is to solve the system of homogeneous linear equations, and the computational effort is large. Using MATLAB, you can easily calculate the eigenvalues and eigenvectors of any complex square matrix

1. First of all, you need to know that the eig function is used to calculate the eigenvalues and eigenvectors of the matrix, you can enter help eig in the command line window to check the usage of the eig function, as shown in the following figure:

2. Enter a 1 2 3 in the command line window2 4 5；7 8 9 , after pressing the enter key, enter x,y eig(a) as shown in the following figure:

3. After pressing the enter key, the value of x,y is obtained, where each column of x represents a feature vector of matrix a, there are 3 eigenvectors, and the diagonal element value of y represents the eigenvalue of matrix a, as shown in the figure below:

4. Step If we want to take the diagonal element value of y, we can use diag(y), as shown in the following figure:

5. After pressing the Enter key, you can see that the value of the diagonal element of the jujube width of Y has been taken out, that is, the eigenvalue of the A matrix, as shown in the following figure:

6. In the sixth step, we can also help diag in the command line window, and you can see the usage of the diag function, as shown in the following figure:

Notes:

Application of eigenvalues and eigenvectors:

1. It can be used in the study of differential equations in the field of physics and chemistry, continuous or discrete dynamical systems. For example, in mechanics, the eigenvectors of inertia define the main axes of rigid bodies. The inertia is the key data that determines the rotation of a rigid body around the center of mass;

2. What extent is the deforestation of primeval forests used by mathematical ecologists to cause the extinction of owl populations;

3. The well-known PCA method in image processing selects K feature vectors with the highest eigenvalues to represent a matrix, so as to achieve dimensionality reduction analysis + feature display method, as well as the K-L transformation of image compression. Another example is a lot of face recognition, data flow pattern mining and analysis, etc.

>>clc;clear;close；>>a=[3,-1,-2;2,0,-2;2,-1,-1]；>x,b]=eig(a) % to find the eigenvalues and eigenvectors of matrix a, where the diagonal element of b is the eigenvalue and the column of %x is the corresponding eigenvector. x = 。

1.First, let's build a square matrix where we need to compute eigenvalues and eigenvectors.

2.Then you need to use the function expressions that come with MATLAB to calculate the eigenvalues and eigenvectors of the square. The format is as follows:

v,d]=eig(a)。

3.Then press the enter key to get the eigenvalues and eigenvectors of the matrix we need to find.