fibonacci_sequence
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fibonacci_sequence [2023/07/26 14:57] – [Eigen vectors of A] raju | fibonacci_sequence [2023/08/05 11:05] (current) – [References] raju | ||
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\left( \lambda_1, \lambda_2 \right) = \left( \frac{1+\sqrt{5}}{2}, | \left( \lambda_1, \lambda_2 \right) = \left( \frac{1+\sqrt{5}}{2}, | ||
$$ | $$ | ||
+ | Also, note | ||
+ | $$ \lambda_1 - \lambda_2 = \sqrt{5} $$ | ||
==== Eigen vectors of A ==== | ==== Eigen vectors of A ==== | ||
Line 62: | Line 64: | ||
\end{bmatrix} | \end{bmatrix} | ||
$$ | $$ | ||
+ | |||
Expanding | Expanding | ||
- | $$ | + | |
\begin{align} | \begin{align} | ||
(1-\lambda) x_1 + x_2 & = 0 \\ | (1-\lambda) x_1 + x_2 & = 0 \\ | ||
x_1 & = \lambda x_2 | x_1 & = \lambda x_2 | ||
\end{align} | \end{align} | ||
- | $$ | ||
From the characteristic equation of $ \mathbf{A} $, we know | From the characteristic equation of $ \mathbf{A} $, we know | ||
- | $$ | + | |
\begin{align} | \begin{align} | ||
& (1-\lambda)(-\lambda) - 1 = 0 \\ | & (1-\lambda)(-\lambda) - 1 = 0 \\ | ||
\Rightarrow \quad & (1-\lambda) = -\frac{1}{\lambda} | \Rightarrow \quad & (1-\lambda) = -\frac{1}{\lambda} | ||
\end{align} | \end{align} | ||
- | $$ | + | |
Substituting for $1-\lambda$, | Substituting for $1-\lambda$, | ||
- | $$ | ||
\begin{align} | \begin{align} | ||
-\frac{1}{\lambda} x_1 + x_2 & = 0 \\ | -\frac{1}{\lambda} x_1 + x_2 & = 0 \\ | ||
x_1 & = \lambda x_2 | x_1 & = \lambda x_2 | ||
\end{align} | \end{align} | ||
- | $$ | ||
So both equations simplify to | So both equations simplify to | ||
$$ x_1 = \lambda x_2$$ | $$ x_1 = \lambda x_2$$ | ||
+ | which gives the eigen vector matrix as | ||
+ | $$ | ||
+ | \Lambda = | ||
+ | \begin{bmatrix} | ||
+ | \lambda_1 & \lambda_2 \\ | ||
+ | 1 & 1 | ||
+ | \end{bmatrix} | ||
+ | $$ | ||
+ | |||
+ | |||
+ | ==== References ==== | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * tags | big parentheses | ||
fibonacci_sequence.1690383472.txt.gz · Last modified: 2023/07/26 14:57 by raju