What is the concept of eigenvalue of matrix in linear algebra

What is the concept of eigenvalue of matrix in linear algebra

Let a be a square matrix of order n. if there exists 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 A. the non-zero n-dimensional column vector x is called the eigenvector or eigenvector of matrix A which belongs to (corresponds to) the eigenvalue m, or the eigenvector of a for short

Do all matrices have eigenvalues

There may be no real eigenvalues, but there must be complex eigenvalues
The reason is that the characteristic polynomial of a matrix can be decomposed into a factor in the complex field, but not in the real field

Who uses MATLAB to help me calculate the maximum eigenvalue of the matrix and the corresponding eigenvector
A=(1,2,3,4,5;0.5,1,2,3,4;0.33,0.5,1,2,3;0.25,0.33,0.25,1,2;0.2,0.25,0.33,0.5,1)

[v,d]=eig(A)v =0.7900 0.8197 0.8197 0.7930 0.7930 0.4940 0.1839 + 0.3933i 0.1839 - 0.3933i -0.3667 + 0.2225i -0.3667 - 0.2225i0.3000 -0.1541 + 0.2270i -0.1541 - 0.2270i -0.0853 - 0.2230i -0.0853 + 0.2...