In [1]:

pip install nacal

Requirement already satisfied: nacal in /home/marcos/.local/lib/python3.9/site-packages (0.1.13)
Requirement already satisfied: sympy>=1.1.0 in /usr/lib/python3/dist-packages (from nacal) (1.7.1)
Note: you may need to restart the kernel to use updated packages.

In [2]:

from nacal import *
from IPython.display import display, Math

$ \renewcommand{\R}[1][{}]{{\mathbb{R}}^{#1}} \renewcommand{\Z}[1][{}]{{\mathbb{Z}}^{#1}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\getitemR}[1]{\getItem{#1}} \renewcommand{\getItem}{\pmb{\mid}} \renewcommand{\elemR}[2]{{#1}^{\phantom{\T}}_{\getitemR{#2}}} \renewcommand{\elemRP}[2]{{\big(#1\big)}^{\phantom{\T}}_{\getitemR{#2}}} \renewcommand{\elemRPE}[2]{\big({#1}^{\phantom{\T}}_{\getitemR{#2}}\big)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\Vect}[2][{}]{{\boldsymbol{#2}}_{#1}} \renewcommand{\eleVR}[2] {\elemR {\Vect{#1}}{#2}} % con subindices \renewcommand{\eleVRP}[2] {\elemRP {\Vect{#1}}{#2}} % con subindices y paréntesis interior \renewcommand{\eleVRPE}[2]{\elemRPE{\Vect{#1}}{#2}} % con subindices y paréntesis exterior %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\VectC}[2][{}] {\elemR {\Mat{#2}}{#1}} % con subindices \renewcommand{\VectCP}[2][{}] {\elemRP {\Mat{#2}}{#1}} % con subindices y paréntesis \renewcommand{\VectCPE}[2][{}]{\elemRPE{\Mat{#2}}{#1}} % con subindices y paréntesis exterior %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\mat}[1]{\mathbf{#1}} \renewcommand{\Mat} [2][{}]{{\mat{#2}}_{#1}} \renewcommand{\T}{\intercal} \renewcommand{\MatT}[2][{}]{{\mat{#2}}^{\T}_{#1}} \renewcommand{\VectC}[2][{}] {\elemR {\Mat{#2}}{#1}} % con subindices \renewcommand{\VectCP}[2][{}] {\elemRP {\Mat{#2}}{#1}} % con subindices y paréntesis \renewcommand{\VectCPE}[2][{}] {\elemRPE {\Mat{#2}}{#1}} % con subindices y paréntesis exterior \renewcommand{\VectCC}[2][{}] {\elemRR {\Mat{#2}}{#1}} % con () \renewcommand{\VectCCC}[2][{}] {\elemRRR{\Mat{#2}}{#1}} % con texto "col" %SELECCIÓNA de FILAS y COlUMNAS DE UNA MATRIZ TRANSPUESTA PARA GENERAR UN VECTOR DE Rn \renewcommand{\VectTC}[2][{}] {\elemR{\MatT{#2}\!}{#1}} % con subindices \renewcommand{\VectTCC}[2][{}] {\elemRR{ \MatT{#2}}{#1}} % con () \renewcommand{\VectTCCC}[2][{}] {\elemRRR{\MatT{#2}}{#1}} % con texto "col" $

${\huge\text{Jupyter notebook de la Lección 17}}$

Índice de contenidos de este notebook (Requiere ejecutar todo el Notebook para que aparezca el índice)

Autovalores y autovectores¶

Considere la siguiente matriz cuadrada

In [3]:

A=Matrix([[0,1],[1,0]])
A

Out[3]:

$\left[ \begin{array}{cc}0&1\\1&0\\ \end{array} \right]$

In [4]:

x=Vector( [1, 1] )
x

Out[4]:

$\begin{pmatrix}1\\1\end{pmatrix}$

In [5]:

A*x

Out[5]:

$\begin{pmatrix}1\\1\end{pmatrix}$

In [6]:

y=Vector( [-5, 5 ] ) #(lambda = -1)
y

Out[6]:

$\begin{pmatrix}-5\\5\end{pmatrix}$

In [7]:

A*y

Out[7]:

$\begin{pmatrix}5\\-5\end{pmatrix}$

In [8]:

A=Matrix([[3,1],[1,3]])
A

Out[8]:

$\left[ \begin{array}{cc}3&1\\1&3\\ \end{array} \right]$

In [9]:

x=Vector( [1, 1] ) # lambda=4
x

Out[9]:

$\begin{pmatrix}1\\1\end{pmatrix}$

In [10]:

A*x

Out[10]:

$\begin{pmatrix}4\\4\end{pmatrix}$

In [11]:

y=Vector( [-1, 1] ) # lambda=2
y

Out[11]:

$\begin{pmatrix}-1\\1\end{pmatrix}$

In [12]:

A*y

Out[12]:

$\begin{pmatrix}-2\\2\end{pmatrix}$

In [13]:

Q = Matrix( [ [0, -1], [1, 0] ] )
Q

Out[13]:

$\left[ \begin{array}{cc}0&-1\\1&0\\ \end{array} \right]$

In [14]:

x = Vector( [1, 2])
x

Out[14]:

$\begin{pmatrix}1\\2\end{pmatrix}$

In [15]:

z=Q*x
z

Out[15]:

$\begin{pmatrix}-2\\1\end{pmatrix}$

In [16]:

Q*Q*Q*Q*x

Out[16]:

$\begin{pmatrix}1\\2\end{pmatrix}$

In [17]:

S = Sistema([ (Q**n)*x for n in range(0,13) ])
S

Out[17]:

$\left[\begin{pmatrix}1\\2\end{pmatrix};\;\begin{pmatrix}-2\\1\end{pmatrix};\;\begin{pmatrix}-1\\-2\end{pmatrix};\;\begin{pmatrix}2\\-1\end{pmatrix};\;\begin{pmatrix}1\\2\end{pmatrix};\;\begin{pmatrix}-2\\1\end{pmatrix};\;\begin{pmatrix}-1\\-2\end{pmatrix};\;\begin{pmatrix}2\\-1\end{pmatrix};\;\begin{pmatrix}1\\2\end{pmatrix};\;\begin{pmatrix}-2\\1\end{pmatrix};\;\begin{pmatrix}-1\\-2\end{pmatrix};\;\begin{pmatrix}2\\-1\end{pmatrix};\;\begin{pmatrix}1\\2\end{pmatrix};\right]$

In [18]:

[ (S|i) * (S|(i+1)) for i in range(1,13) ]

Out[18]:

[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

In [19]:

A = Matrix( [[3, 1], [0, 3]] )
A

Out[19]:

$\left[ \begin{array}{cc}3&1\\0&3\\ \end{array} \right]$

In [20]:

Homogenea(A-3*I(2),1)

$\displaystyle \left[ \begin{array}{cc}0&1\\0&0\\ \hline 1&0\\0&1\\ \end{array} \right]$

Out[20]:

$\mathcal{L}\left(\ \left[\begin{pmatrix}1\\0\end{pmatrix};\right]\ \right)$

In [21]:

x = Vector( [1, 0] )
x

Out[21]:

$\begin{pmatrix}1\\0\end{pmatrix}$

In [22]:

A*x

Out[22]:

$\begin{pmatrix}3\\0\end{pmatrix}$

Ejercicio 1

In [23]:

A=Matrix([[-3,4,-4], [-3,5,-3], [-1,2,0]])
A

Out[23]:

$\left[ \begin{array}{ccc}-3&4&-4\\-3&5&-3\\-1&2&0\\ \end{array} \right]$

In [24]:

m = Vector([0,1,1]);m

Out[24]:

$\begin{pmatrix}0\\1\\1\end{pmatrix}$

In [25]:

A*m

Out[25]:

$\begin{pmatrix}0\\2\\2\end{pmatrix}$

In [26]:

v= Vector([2,1,0]); v

Out[26]:

$\begin{pmatrix}2\\1\\0\end{pmatrix}$

In [27]:

A*v

Out[27]:

$\begin{pmatrix}-2\\-1\\0\end{pmatrix}$

In [28]:

-1*v

Out[28]:

$\begin{pmatrix}-2\\-1\\0\end{pmatrix}$

In [29]:

w=Vector([1,0,-1]);w

Out[29]:

$\begin{pmatrix}1\\0\\-1\end{pmatrix}$

In [30]:

A*w

Out[30]:

$\begin{pmatrix}1\\0\\-1\end{pmatrix}$

In [31]:

1*w

Out[31]:

$\begin{pmatrix}1\\0\\-1\end{pmatrix}$

In [32]:

A-2*I(3)

Out[32]:

$\left[ \begin{array}{ccc}-5&4&-4\\-3&3&-3\\-1&2&-2\\ \end{array} \right]$

In [33]:

Homogenea(A-2*I(3),1)

$\displaystyle \left[ \begin{array}{ccc}-5&4&-4\\-3&3&-3\\-1&2&-2\\ \hline 1&0&0\\0&1&0\\0&0&1\\ \end{array} \right]\xrightarrow{\underset{\begin{subarray}{c} \left[\left(5\right)\mathbf{2}\right]\\\left[\left(4\right)\mathbf{1}+\mathbf{2} \right]\\\left[\left(5\right)\mathbf{3}\right]\\\left[\left(-4\right)\mathbf{1}+\mathbf{3} \right]\end{subarray}}{\pmb{\tau}}}\left[ \begin{array}{ccc}-5&0&0\\-3&3&-3\\-1&6&-6\\ \hline 1&4&-4\\0&5&0\\0&0&5\\ \end{array} \right]\xrightarrow{\underset{\begin{subarray}{c} \left[\left(1\right)\mathbf{2}+\mathbf{3} \right]\end{subarray}}{\pmb{\tau}}}\left[ \begin{array}{ccc}-5&0&0\\-3&3&0\\-1&6&0\\ \hline 1&4&0\\0&5&5\\0&0&5\\ \end{array} \right]$

Out[33]:

$\mathcal{L}\left(\ \left[\begin{pmatrix}0\\5\\5\end{pmatrix};\right]\ \right)$

---- FIN DE LA LECCIÓN -----

Lo que sigue es sólo para crear un índice de contenidos de este notebook (y se muestra al principio de la página)

In [34]:

%%javascript
$.getScript('https://kmahelona.github.io/ipython_notebook_goodies/ipython_notebook_toc.js')
$.getScript('https://kmahelona.github.io/ipython_notebook_goodies/ipython_notebook_toc.js')