interpreting eigenvectors/values of linear systems

Question

I have two related questions:

1. If I have a system of linear equations, written in matrix form:

   $Ax = b$

   then this system has a non-trivial solution only if $det(A) \neq 0$. but what is the significance of the eigenvalues/eigenvectors of A for finding the solutions?

2. Related, if I have a system of homogeneous ordinary differential equations:

   $x' = Ax$

   then what is the interpretation of eigenvalues/eigenvectors of A in finding the solutions of this system of equations?

pointers to explanations or concise explanations to this would help.

Answer 1

Considering eigenvectors this should imply quadratic matrices.

Solving $Ax=b$ then means $x=A^{-1}b$. If we have a base of eigenvectors, $A$ has $n$ distinct eigenvectors, we have $$ \DeclareMathOperator{diag}{diag} b = A x = \diag(\lambda_1, \dotsc, \lambda_n) x $$ where $diag$ is a diagonal matrix, and the invertation simplifies to $$ x = \diag(\lambda_1^{-1},\dotsc,\lambda_n^{-1}) b $$

For the linear ODE system, we have the solution $$ x(t) = e^{tA} x(0) $$ Here the matrix exponential simplifies to $$ e^{tA} = \diag(e^{\lambda_1 t}, \dotsc, e^{\lambda_n t}) $$