Why eigenvalues

Eigenvalues turn a matrix multiplication into a multiplication by a scalar:

$$Av=\lambda v$$

Diagonalizable matrices have a full eigenvector basis and can be written in the form:

$$A=X\Lambda X^{-1}$$

In this form, taking the power of a matrix is a simple operation:

$$A^k=X{\Lambda }^k{X}^{-1}$$

We can even compute the function of a matrix by using the eigendecomposition:

$$f(A)=Xf(\Lambda )X^{-1}$$

where $f(\Lambda )$ is a diagonal matrix with entries $f({\lambda }_i)$.

Eigenvalues are very useful to study time-evolving systems. Consider

$$\frac{dx}{dt}=Mx$$

We can formally write the solution in the form:

$$x(t)=\exp (Mt)x_0$$

where $\exp (Mt)$ is defined as above using functions of matrices:

$$\exp (Mt)=X\exp (\Lambda t)X^{-1}$$

with

$$M=X\Lambda X^{-1}$$

and where $\exp (\Lambda t)$ is a diagonal matrix with entries $\exp ({\lambda }_{i}t)$.

This provides an exact solution to the linear system $\dot{x}=Mx$.