-chaper5-eigenvalues and eigenvectors of Linear Algebra and its applications

Based on the previous chapters, we can easily draw the concept of eigenvectors and eigenvalues.

First we know that the product of a and n dimensional vector v of n x n matrices will get an n-dimensional vector, then we now find that, after calculating u=av, the resulting vector u is collinear with V, that is, vector v is multiplied by matrix A to get the vector u "stretched" with respect to vector V, which satisfies the following equation:

Av =λv=u

So here we call λ the eigenvalues of matrix A, and V is the characteristic vector of the corresponding eigenvalues.

The rigor is defined as follows:

Theorem 1:

The element of the triangular matrix's main diagonal is its eigenvalues.

Before we prove it, we first need to do a more thorough digging of the definition, the eigenvector x cannot be a 0 vector, and we will change the formula in the definition, namely:

Matrix equation (a-λi) x=0, when there is a non-trivial solution, there is a characteristic value λ exists.

-chaper5-eigenvalues and eigenvectors of Linear Algebra and its applications