Eigenvalues and eigenvectors (eigenvalues and eigenvectors)

Eigenvalues and eigenvectors are the essential contents of matrices, which play an important role in dynamic problems, and the matrix is the square (square) By default in this paper. 1. Geometrical significance

Now we explain from a geometrical point of view that the eigenvalues are what the eigenvectors are. When most vectors (x) multiply by the matrix A, that is, Ax, (as mentioned in the following vector x by a-value ax) changes the direction of the vector, but there is a matrix x outside some columns, which is in the same direction as AX, and these vectors are called eigenvectors . Vector ax is a scalar multiply on the original vector x.

The size of the eigenvalues indicates that the feature vector x is stretched, shrunk, reversed, or not changed when it is multiplied by a (e.g. The corresponding value is 2,-1/2,-1,1). Of course, the eigenvalues can also be 0. Indicates that the eigenvector is in the 0 space of the matrix. If a is a unit matrix, all vectors are satisfied, all vectors are eigenvectors, and all eigenvalues are 1.

2. The solution of eigenvalue eigenvectorEigenvalue is solved by equation, and the corresponding eigenvalue is 0 space after finding the eigenvalues. The following example is the process of solving:




The following describes a simple use of eigenvalues, we can find that if a by X1, we get x1, and so on, the same a^n*x2 = (a) ^n * x2, from which we can get the same feature vector as X1,X2, and the eigenvalues have changed, respectively, 1 and.
Because the eigenvectors of different eigenvalues are linearly independent, all X1 and X2 can be used as a set of base vectors for two-dimensional vectors, and all two-dimensional vectors can be represented as X1 and X2 thread combinations. We can do it. The first column vector of matrix A is decomposed into:
When you multiply the matrix A, you get:

Get the result (. 7,. 3) as the first column vector for a^2. Of course we need to solve the matrix, you can use the same method, the following is the first column by solving the vector:

According to the method of appeal, when we need to solve the higher power of a matrix A, I do not need to ask a^2,a^3 ..., such efficiency is very low, we can directly through the eigenvalues and eigenvectors to solve. The above mentioned X1 will not change, we call it "steady state", because his eigenvalues are 1,x2 will slowly disappear, we become "decaying mode" because its eigenvalues are less than 1. According to this property, each column of a matrix with a higher power tends to be steady state. Here we introduce the Markov matrix, which all elements are positive, each column is 1, and its maximum eigenvalue is 1. The matrix A above is the Markov matrix.

3. Several properties1. If the and of each column of matrix A is 1, then 1 is a characteristic value of a. 2. If the matrix A is singular, det (a) = 0, then 0 is a characteristic value of a. 3. If a is a symmetric matrix, the eigenvectors of different eigenvalues are orthogonal to each other. 4. The product of all eigenvalue values is the determinant of the matrix. 5. The and of all eigenvalues are the traces of the matrix, that is, the elements on the main diagonal of the matrix.
4. SummaryIn this paper, we briefly introduce the eigenvalues and eigenvectors, the above examples are more ideal (a matrix has n linearly independent eigenvalues). It should be noted that some n*n matrices do not have independent eigenvectors of N, so it is not possible to be a base of n-dimensional space, and the same cannot represent all n-dimensional vectors. (such matrices are not likely to be diagonal)
5.Reference"A Introduction to Linear Algebra" GILBERT Strang