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"\n",
"\n",
"\n",
"*This notebook contains an excerpt from the [Python Programming and Numerical Methods - A Guide for Engineers and Scientists](https://www.elsevier.com/books/python-programming-and-numerical-methods/kong/978-0-12-819549-9), the content is also available at [Berkeley Python Numerical Methods](https://pythonnumericalmethods.berkeley.edu/notebooks/Index.html).*\n",
"\n",
"*The copyright of the book belongs to Elsevier. We also have this interactive book online for a better learning experience. The code is released under the [MIT license](https://opensource.org/licenses/MIT). If you find this content useful, please consider supporting the work on [Elsevier](https://www.elsevier.com/books/python-programming-and-numerical-methods/kong/978-0-12-819549-9) or [Amazon](https://www.amazon.com/Python-Programming-Numerical-Methods-Scientists/dp/0128195495/ref=sr_1_1?dchild=1&keywords=Python+Programming+and+Numerical+Methods+-+A+Guide+for+Engineers+and+Scientists&qid=1604761352&sr=8-1)!*"
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"\n",
"< [CHAPTER 15. Eigenvalues and Eigenvectors](chapter15.00-Eigenvalues-and-Eigenvectors.ipynb) | [Contents](Index.ipynb) | [15.2 The Power Method](chapter15.02-The-Power-Method.ipynb) >"
]
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"source": [
"# Eigenvalues and Eigenvectors Problem Statement"
]
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"## Eigenvalues and eigenvectors \n",
"\n",
"We learned from last chapter that matrix $A$ apply to column vector $x$, that is $Ax$, is a linear transformation of $x$. There is a special transform in the following form: \n",
"\n",
"$$Ax = \\lambda{x}$$\n",
"\n",
"Where $A$ is $n\\times{n}$ matrix, $x$ is $n\\times{1}$ column vector ($X\\neq{0}$), and $\\lambda$ is some scalar. Any $\\lambda$ that satisfies the above equation is known as an **eigenvalue** of the matrix $A$, while the associated vector $x$ is called an **eigenvector** corresponding to $\\lambda$. \n",
"\n",
"## The motivation behind\n",
"\n",
"The motivation behind the eigenvalues and eigenvectors is that, it helps us to understand the characteristics of the linear transformation, thus make things easy. We know that a vector $x$ can be transformed to a different vector by multiplying $A$ - $Ax$. The effect of the transformation represents a scale of the length of the vector and/or the rotate of the vector. The above equation points out that for some vectors, the effect of transformation of $Ax$ is only scale (stretching, compressing, and flipping). The eigenvectors are the vectors have this property and the eigenvalues $\\lambda's$ are the scale factors. Let's look at the following example. \n",
"\n",
"**TRY IT** Plot the vector $x$ = [[1], [1]] and the vector $b = Ax$, where $A$ = [[2, 0], [0, 1]]"
]
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{
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"execution_count": 1,
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"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"\n",
"plt.style.use('seaborn-poster')\n",
"\n",
"%matplotlib inline\n",
"\n",
"def plot_vect(x, b, xlim, ylim):\n",
" '''\n",
" function to plot two vectors, \n",
" x - the original vector\n",
" b - the transformed vector\n",
" xlim - the limit for x\n",
" ylim - the limit for y\n",
" '''\n",
" plt.figure(figsize = (10, 6))\n",
" plt.quiver(0,0,x[0],x[1],\\\n",
" color='k',angles='xy',\\\n",
" scale_units='xy',scale=1,\\\n",
" label='Original vector')\n",
" plt.quiver(0,0,b[0],b[1],\\\n",
" color='g',angles='xy',\\\n",
" scale_units='xy',scale=1,\\\n",
" label ='Transformed vector')\n",
" plt.xlim(xlim)\n",
" plt.ylim(ylim)\n",
" plt.xlabel('X')\n",
" plt.ylabel('Y')\n",
" plt.legend()\n",
" plt.show()"
]
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\n",
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