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--- fibonacci_sequence [2023/07/26 13:56] – raju
+++ fibonacci_sequence [2023/08/05 11:05] (current) – [References] raju
@@ Line 23: / Line 23: @@
 ==== Eigen values of A ====
+To get the eigen values, solve
+$$ \begin{vmatrix}
+- \lambda & 1 \\
+& -\lambda
+\end{vmatrix} = 0
+$$
+\begin{align}
+(1-\lambda)(-\lambda) - 1 & = 0 \\
+(\lambda)(\lambda - 1) - 1 & = 0 \\
+\lambda^2 - \lambda -1 & = 0
+\end{align}
+If we denote the eigen values as $ \lambda_1, \lambda_2 $, this tells us that
+$$ \lambda_1 + \lambda_2 = 1, \quad \lambda_1 \lambda_2 = -1 $$
+Solving the quadratic equation, we get
+$$
+\left( \lambda_1, \lambda_2 \right) = \left( \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2} \right)
+$$
+Also, note
+$$ \lambda_1 - \lambda_2 = \sqrt{5} $$
+==== Eigen vectors of A ====
+To get the eigen vectors, solve
+$$
+\begin{bmatrix}
+- \lambda & 1 \\
+& -\lambda
+\end{bmatrix}
+\begin{bmatrix}
+x_1 \\
+x_2
+\end{bmatrix}
+=
+\begin{bmatrix}
+\\
+\end{bmatrix}
+$$
+Expanding
+\begin{align}
+(1-\lambda) x_1 + x_2 & = 0 \\
+x_1 & = \lambda x_2
+\end{align}
+From the characteristic equation of $ \mathbf{A} $, we know
+\begin{align}
+& (1-\lambda)(-\lambda) - 1 = 0 \\
+\Rightarrow \quad & (1-\lambda) = -\frac{1}{\lambda}
+\end{align}
+Substituting for $1-\lambda$, we get
+\begin{align}
+-\frac{1}{\lambda} x_1 + x_2 & = 0 \\
+x_1 & = \lambda x_2
+\end{align}
+So both equations simplify to
+$$ x_1 = \lambda x_2$$
+which gives the eigen vector matrix as
+$$
+\Lambda =
+\begin{bmatrix}
+\lambda_1 & \lambda_2 \\
+& 1
+\end{bmatrix}
+$$
+==== References ====
+  * https://www.dokuwiki.org/plugin:mathjax - mathjax syntax of dokuwiki
+  * https://tex.stackexchange.com/questions/162533/how-to-align-implies-that-symbols-neatly-in-equation-array - talks about alignat which can be used to align 'implies that' symbols in an equation array. I picked up ''\Rightarrow\quad'' from here.
+  * https://mathworld.wolfram.com/CharacteristicEquation.html
+  * https://www.math-linux.com/latex-26/faq/latex-faq/article/how-to-write-matrices-in-latex-matrix-pmatrix-bmatrix-vmatrix-vmatrix - nice visual representation of how various matrix environments work in latex
+  * https://tex.stackexchange.com/questions/78736/bigger-parentheses-in-equations - picked up ''\left( ... \right)'' from here.
+    * tags | big parentheses