While exploring the relationship between characteristic equations and recurrence relations in linear algebra, I discovered their connection to the Fibonacci sequence.
Let's say there is a recurrence relation $a_n = a_{n-1} + a_{n-2}$. The order of this recurrence relation is $2$. Substituting this into a new framework $x^n$, we get $x^n = x^{n-1} +x^{n-2}$, and restricting $n$ by order, we get $x^2 = x+1$. Now, if we rearrange this equation into binomial form $x^2-x-1=0$, the solutions to this equation are $x=\frac{1+\sqrt{5}}{2} , \frac{1-\sqrt{5}}{2}$. A linear combination of a specific constant multiple of these two solutions becomes the general term of this recurrence relation. So, $a_n = k_1 \times x_{1}^{n} + k_2 \times x_{2}^{n}$ is obtained, and if we find a specific constant multiple $(k_1, k_2)$ through the initial conditions, the general term of the above recurrence relation is $$a_n = \frac{1}{\sqrt{5}} \times (\frac {1+\sqrt{5}}{2})^n - \frac{1}{\sqrt{5}} \times (\frac {1-\sqrt{5}}{2})^n,$$ which is the general term of the Fibonacci sequence.
Furthermore, the decomposition theorem emerges during the process of factoring and connecting characteristic equations. Then, if we briefly turn our attention to eigenvectors, we see that in this branch, eigenvalues overlap, leading to a shortage of eigenvectors. Therefore, we can transform to Jordan form, a process similar to diagonalization.
As I studied linear algebra, I realized that the concepts of eigenvectors, diagonalization, decomposition theorem, and Jordan form all stem from a single narrative.
Therefore, I want to elaborate on this story and understand the rationale and motivation behind these developments.
