Understanding the nonlinear relationship between eigenvectors of different eigenvalues

Understanding the nonlinear relationship between eigenvectors of different eigenvalues

I want to try to understand the theorem in a simple and intuitive way, avoiding the derivation of formulas, believing that a lot of people like me feel that abstract mathematical formulas are difficult to remember: mathematical formula characteristic equation:

Ax⃗=λx⃗a \vec{x} = \lambda \vec{x}
The above characteristic equation can be understood as vector x⃗\vec{x} after the linear transformation a a A, the direction is unchanged, the length of the vector x⃗\vec{x} to do the scaling transformation, the corresponding eigenvalue λ\lambda for the direction of the scaling factor parameter.

theorem Understanding

It is assumed that the existence of a constant quantity can be represented by a linear combination of other eigenvectors, i.e.
xt→=∑i=0nkixi→\vec{x_t} = \sum_{i=0}^n k_i \vec {x_i}
Only if xt→\vec{x_t} has the same transformation rate in each xi→ (ki≠0) \vec {x_i} (k_i \not =0) direction after the linear transformation, the direction of xt→\vec{x_t} does not change, and the transformation rate λt=λi (ki≠0) \lam BDA _t = \lambda _i (k_i \not =0). Otherwise, the direction of the vector changes because the rate of transformation in each direction is inconsistent.