fibonacci_sequence
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fibonacci_sequence [2023/07/26 14:28] – [Eigen values 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 ==== | ||
- | To get the eigen vectors solve | + | To get the eigen vectors, solve |
$$ | $$ | ||
\begin{bmatrix} | \begin{bmatrix} | ||
Line 63: | Line 65: | ||
$$ | $$ | ||
+ | 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 $1-\lambda$, | ||
+ | |||
+ | \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 | ||
+ | $$ | ||
+ | \Lambda = | ||
+ | \begin{bmatrix} | ||
+ | \lambda_1 & \lambda_2 \\ | ||
+ | 1 & 1 | ||
+ | \end{bmatrix} | ||
+ | $$ | ||
+ | |||
+ | |||
+ | ==== References ==== | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * https:// | ||
+ | * tags | big parentheses | ||
fibonacci_sequence.1690381735.txt.gz · Last modified: 2023/07/26 14:28 by raju