{ "cells": [ { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "from sympy import *\n", "import numpy as np\n", "init_printing()" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[-2, 1, 0, 0],\n", " [ 1, -2, 1, 0],\n", " [ 0, 1, -2, 1],\n", " [ 0, 0, 1, -2]])" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = np.array([[-2,1,0,0],[1,-2,1,0],[0,1,-2,1],[0,0,1,-2]])\n", "a" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [], "source": [ "from numpy.linalg import eig, det" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "evals, evects = eig(a)" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([ 0.37174803, -0.60150096, 0.60150096, -0.37174803])" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "evects[:,0]" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 0.37174803, 0.60150096, -0.37174803, -0.60150096],\n", " [-0.60150096, -0.37174803, -0.60150096, -0.37174803],\n", " [ 0.60150096, -0.37174803, -0.60150096, 0.37174803],\n", " [-0.37174803, 0.60150096, -0.37174803, 0.60150096]])" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "evects" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle -1.0$" ], "text/plain": [ "-0.9999999999999999" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "det(evects)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "array([[ 3, 5, -4],\n", " [-3, -2, 4],\n", " [ 6, 1, -8]])" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = np.array([[3,5,-4], [-3, -2, 4], [6,1,-8]], np.int64)\n", "a\n" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "th = symbols(\"theta\", real=True)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\theta \\right)} & - \\sin{\\left(\\theta \\right)}\\\\\\sin{\\left(\\theta \\right)} & \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$" ], "text/plain": [ "⎡cos(θ) -sin(θ)⎤\n", "⎢ ⎥\n", "⎣sin(θ) cos(θ) ⎦" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = Matrix(((cos(th), -sin(th) ),(sin(th), cos(th))))\n", "m" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[ \\left( - \\sqrt{\\left(\\cos{\\left(\\theta \\right)} - 1\\right) \\left(\\cos{\\left(\\theta \\right)} + 1\\right)} + \\cos{\\left(\\theta \\right)}, \\ 1, \\ \\left[ \\left[\\begin{matrix}\\frac{\\sin{\\left(\\theta \\right)}}{\\sqrt{\\cos^{2}{\\left(\\theta \\right)} - 1}}\\\\1\\end{matrix}\\right]\\right]\\right), \\ \\left( \\sqrt{\\left(\\cos{\\left(\\theta \\right)} - 1\\right) \\left(\\cos{\\left(\\theta \\right)} + 1\\right)} + \\cos{\\left(\\theta \\right)}, \\ 1, \\ \\left[ \\left[\\begin{matrix}- \\frac{\\sin{\\left(\\theta \\right)}}{\\sqrt{\\cos^{2}{\\left(\\theta \\right)} - 1}}\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$" ], "text/plain": [ "⎡⎛ ⎡⎡ sin(θ) ⎤⎤⎞ ⎛ \n", "⎢⎜ ⎢⎢────────────────⎥⎥⎟ ⎜ \n", "⎢⎜ ___________________________ ⎢⎢ _____________⎥⎥⎟ ⎜ _____\n", "⎢⎜- ╲╱ (cos(θ) - 1)⋅(cos(θ) + 1) + cos(θ), 1, ⎢⎢ ╱ 2 ⎥⎥⎟, ⎜╲╱ (cos\n", "⎢⎜ ⎢⎢╲╱ cos (θ) - 1 ⎥⎥⎟ ⎜ \n", "⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ \n", "⎣⎝ ⎣⎣ 1 ⎦⎦⎠ ⎝ \n", "\n", " ⎡⎡ -sin(θ) ⎤⎤⎞⎤\n", " ⎢⎢────────────────⎥⎥⎟⎥\n", "______________________ ⎢⎢ _____________⎥⎥⎟⎥\n", "(θ) - 1)⋅(cos(θ) + 1) + cos(θ), 1, ⎢⎢ ╱ 2 ⎥⎥⎟⎥\n", " ⎢⎢╲╱ cos (θ) - 1 ⎥⎥⎟⎥\n", " ⎢⎢ ⎥⎥⎟⎥\n", " ⎣⎣ 1 ⎦⎦⎠⎦" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m.eigenvects()" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left( \\left[\\begin{matrix}- \\frac{i \\sin{\\left(\\theta \\right)}}{\\left|{\\sin{\\left(\\theta \\right)}}\\right|} & \\frac{i \\sin{\\left(\\theta \\right)}}{\\left|{\\sin{\\left(\\theta \\right)}}\\right|}\\\\1 & 1\\end{matrix}\\right], \\ \\left[\\begin{matrix}- \\sqrt{\\left(\\cos{\\left(\\theta \\right)} - 1\\right) \\left(\\cos{\\left(\\theta \\right)} + 1\\right)} + \\cos{\\left(\\theta \\right)} & 0\\\\0 & \\sqrt{\\left(\\cos{\\left(\\theta \\right)} - 1\\right) \\left(\\cos{\\left(\\theta \\right)} + 1\\right)} + \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]\\right)$" ], "text/plain": [ "⎛⎡-ⅈ⋅sin(θ) ⅈ⋅sin(θ)⎤ ⎡ ___________________________ \n", "⎜⎢────────── ────────⎥ ⎢- ╲╱ (cos(θ) - 1)⋅(cos(θ) + 1) + cos(θ) \n", "⎜⎢ │sin(θ)│ │sin(θ)│⎥, ⎢ \n", "⎜⎢ ⎥ ⎢ ________\n", "⎝⎣ 1 1 ⎦ ⎣ 0 ╲╱ (cos(θ)\n", "\n", " ⎤⎞\n", " 0 ⎥⎟\n", " ⎥⎟\n", "___________________ ⎥⎟\n", " - 1)⋅(cos(θ) + 1) + cos(θ)⎦⎠" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m.diagonalize()" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left ( \\left[\\begin{matrix}1 & 0 & - \\frac{4}{3}\\\\0 & 1 & 0\\\\0 & 0 & 0\\end{matrix}\\right], \\quad \\left ( 0, \\quad 1\\right )\\right )$$" ], "text/plain": [ "⎛⎡1 0 -4/3⎤ ⎞\n", "⎜⎢ ⎥ ⎟\n", "⎜⎢0 1 0 ⎥, (0, 1)⎟\n", "⎜⎢ ⎥ ⎟\n", "⎝⎣0 0 0 ⎦ ⎠" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "Matrix(a).rref()" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "r1, r2, c, c1, c2 = symbols(\"R_1 R_2 C C_1 C_2\")" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- \\frac{1}{C_{1} R_{1}} & \\frac{1}{C_{2} R_{1}}\\\\\\frac{1}{C_{1} R_{2}} & - \\frac{1}{R_{2} \\left(C_{1} + C_{2}\\right)}\\end{matrix}\\right]$" ], "text/plain": [ "⎡ -1 1 ⎤\n", "⎢───── ───── ⎥\n", "⎢C₁⋅R₁ C₂⋅R₁ ⎥\n", "⎢ ⎥\n", "⎢ 1 -1 ⎥\n", "⎢───── ────────────⎥\n", "⎣C₁⋅R₂ R₂⋅(C₁ + C₂)⎦" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = Matrix(((-1/(r1*c1), 1/(r1*c2)), (1/(r2*c1), -1/(r2*(c1+c2))))); m" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "eigens = m.eigenvects()" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "data": { "image/png": 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hWQa8BLwFPAPs1XL9fbBxf/ellgUV22jCrhA05ZsQYvLgk8tC5KBYc3wI23xRjhdCCCGEEEIIIYQQQgghhBBCCCGEEEIIIYQQQgghxITif2o3LcHgeFPoAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle - \\frac{C_{1} R_{1} + C_{1} R_{2} + C_{2} R_{2}}{2 C_{1} R_{1} R_{2} \\left(C_{1} + C_{2}\\right)} - \\frac{\\sqrt{C_{2} \\left(4 C_{1}^{3} R_{1} R_{2} + C_{1}^{2} C_{2} R_{1}^{2} + 6 C_{1}^{2} C_{2} R_{1} R_{2} + C_{1}^{2} C_{2} R_{2}^{2} + 2 C_{1} C_{2}^{2} R_{1} R_{2} + 2 C_{1} C_{2}^{2} R_{2}^{2} + C_{2}^{3} R_{2}^{2}\\right)}}{2 C_{1} C_{2} R_{1} R_{2} \\left(C_{1} + C_{2}\\right)}$" ], "text/plain": [ " _________________________________________________\n", " ╱ ⎛ 3 2 2 2 \n", " C₁⋅R₁ + C₁⋅R₂ + C₂⋅R₂ ╲╱ C₂⋅⎝4⋅C₁ ⋅R₁⋅R₂ + C₁ ⋅C₂⋅R₁ + 6⋅C₁ ⋅C₂⋅R₁⋅R₂ + \n", "- ───────────────────── - ────────────────────────────────────────────────────\n", " 2⋅C₁⋅R₁⋅R₂⋅(C₁ + C₂) 2⋅C₁⋅C₂⋅R₁⋅\n", "\n", "______________________________________________________\n", " 2 2 2 2 2 3 2⎞ \n", "C₁ ⋅C₂⋅R₂ + 2⋅C₁⋅C₂ ⋅R₁⋅R₂ + 2⋅C₁⋅C₂ ⋅R₂ + C₂ ⋅R₂ ⎠ \n", "──────────────────────────────────────────────────────\n", "R₂⋅(C₁ + C₂) " ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eigens[0][0]" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\frac{C_{2} \\left(C_{1} R_{1} + C_{1} R_{2} + C_{2} R_{2}\\right) + \\sqrt{C_{2} \\left(4 C_{1}^{3} R_{1} R_{2} + C_{1}^{2} C_{2} R_{1}^{2} + 6 C_{1}^{2} C_{2} R_{1} R_{2} + C_{1}^{2} C_{2} R_{2}^{2} + 2 C_{1} C_{2}^{2} R_{1} R_{2} + 2 C_{1} C_{2}^{2} R_{2}^{2} + C_{2}^{3} R_{2}^{2}\\right)}}{2 C_{1} C_{2} R_{1} R_{2} \\left(C_{1} + C_{2}\\right)}$" ], "text/plain": [ " ⎛ ____________________________________________\n", " ⎜ ╱ ⎛ 3 2 2 2 \n", "-⎝C₂⋅(C₁⋅R₁ + C₁⋅R₂ + C₂⋅R₂) + ╲╱ C₂⋅⎝4⋅C₁ ⋅R₁⋅R₂ + C₁ ⋅C₂⋅R₁ + 6⋅C₁ ⋅C₂⋅R₁⋅\n", "──────────────────────────────────────────────────────────────────────────────\n", " 2⋅C₁⋅C₂⋅R₁⋅R₂⋅(C₁ + \n", "\n", "___________________________________________________________⎞ \n", " 2 2 2 2 2 3 2⎞ ⎟ \n", "R₂ + C₁ ⋅C₂⋅R₂ + 2⋅C₁⋅C₂ ⋅R₁⋅R₂ + 2⋅C₁⋅C₂ ⋅R₂ + C₂ ⋅R₂ ⎠ ⎠ \n", "─────────────────────────────────────────────────────────────\n", "C₂) " ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(eigens[0][0])" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKcAAAAwCAYAAAB0dWoXAAAABHNCSVQICAgIfAhkiAAABrBJREFUeJztnHuIFVUcxz+7PlZD1NpaNCgVNxV8tL7StOSmUtjLEFF72VpIRVAU0ZMiolIr0tCCssLeL/9Iwyhca4kybJPU7EEkGZlGDyXLatPa/vjOurNzZ+ae2Zl778zu+cBw9545c85vZn9zfr/zO+d3wWKxWFJED+AaoCXNR/ei3b4lzeSA34GKMsthSSGTKO6oU4hHgGMTu5siUVluAbooE9GoVayjEP2BAzHkrwSagLUduPYkoBH4AtgOzAmqaM166ekGHC5j/2OAz2K2cS2wi47pzxHgBqSYNcBW4G3gz5gyWRJgGjC6jP3fCQyNcX0NsAmYTsdGTi87gJP9TlizXnpGE3/kisNQNOp1lIeAu4D/EpBlAoocfO930ipn6fGbsEwD3gT2Oufnes7vxn/i81jEvgcA+0LOnwq8COwB/ga+BZ4HRrrkbAE2h7Txrku+w8DXwJU+9aqB54CrMJvEWYrMMGCGT/ks4D40OfBTzhOQYrUeM516uYj9LwZODzi3EPmDzwBTkak9EynnE06d24Af0MvyI3AIeNrTzgHgdkfOQcADaJQd46pTBbwPXB5RfkuCLATOc31fjMxYGH7K6WUF8A3R45Sr8beWU5Bi3hhw3XE+ZTnyfc6hSP6xrrLBTtkC53sF8DJwTyFhrVkvHlXAT8AlrrIexJ+p9wQuQyOcnznsDjwMzPOU90am2s9XXIFM9fKAPvcbyjYeOIgmOQADgQedPj91yqYC84GLgG3OUc4JohHXoZs66Bwf0X7UySofIIWpJl9h/Cg0cs5Do9yJAednAuuA1z3lFwKzfeqPcPqcbyBbIZYB/wJ/oNBQC9CMQkeZZjZwLnAK8s3uR6PMmLCLMsASFHZZgILfhSiknO+gyVMYtWh5sspVthI4xqfuAqfPWgPZ1iOfMiiE1ID801pgnCPryohtZIb9wNXlFiImk4FHUeDahDDlHIRGJr8R0MvnaKIF8vNWB9Rb5PQ5wKDNs4ALCFasX4ErXN9b5R3pKivUxlHS6nN2Ay4G+hAetsgCW5CCNifQ1iLkx24wqLueNiU+DS03+rHT+ZwWcN492r6HRmQ/hqCJkzuG+x3yKS81bKMdaVPO0chfaQYeR05zOQPWSdCCfOnGkDp9gDrnAM1w62i/clKJlPNZ5HMWYh3yMyuA8wl2BZqAt4BVQD1yq2qRD7oJxT5NGI8mPl96yjcSsn6eJXqiBzMB+Wq/AKPKKlEyVBc4n8M/yL7GVedsp2yYYZ8VKOA+ifxYpJcq4Bb0Eh1CPuFW4F6gl4+sfiZ5CQq4e5nhyO027UFtZIoG4KlyC5FhngReQuvpSZEjvmIZtZH2XUmV5L+5FnPWI3NuapotASxFy2WDke+5BPkws0KusYTTi7YJTxI0AD+jGOYegpdCi91GyVmDZnfNaEbaAJxTToE6CTXlFsBi6fTsJlquygtlkbJ47KZr33+q8E6IdqHNAabsTVCWNNDV799isZiQVN5y1nYylyJfO2vPxGJJBZtR9uNO4O4OXG+c3ptG7kAjx6qA84VyVQphmqdi8aev89kN7ZutC6nrx0DaAvs1KEHNbzteLIqx8WMySkfYEXB+IVq3bUabC4Y5ZQDXG/YxFr0AA9Fa/Fq0zJn1vZ+l4qDz2dM5orIPjZigmPQB4PgE5Coq/dCMdzoa9r0jZ0dyVbyY5Kl0ZpKySlvQ1rVlnvKoVmkC2omU+t9depW2m20k/wF+jLLu4jAP+A2ZJNDo+Rra1Do8ZttpZzJStu34K6dJBqWbvmi3unvnl0n2ZCvVyO+cEv1WSstiZK5bzUQj7R9glFyVsK38JnkqnSYVwEWxrNKtwM3O31GsUmbSe4ejxfwRrrJG2j/AKLkqYVv5TfJUjFMBMkRSVqkfyoMHbQzZhDYjg7lVMk7vTQP1SPGOuI4WZA6OoLcsSq4KBO/5M8lTCbs+iyRplQY7be0gP5Rkmj15BvrfbiMD6b39kd/iPprQRtdR6E2biG42KD3WG4rIka9cQ5w2xnnKtyLfqND1WSRpqxSGiVXqFDSSb3o2oNBDPfm5Kt59fTnylWsuerN7e8qXAl8ZXJ9F6knWKoX546ZWqSQ+fakT3OagX6O4CZmCJpS78iFtvwgRxng0KfjLU74RjTCmQfws8QYymXWu4xPgFefvf4iWQbmctriyG9PsybA2ugw54r2Zca9PM42U1yoFtdEliLuVP5OpABFoJF8542ZQRsmeDGrDYolNjhJlUMYhbT+qYLEcxSqnxWJJDV02vddisVgsFovFYvHyP6A3Lgx6AJ6pAAAAAElFTkSuQmCC\n", "text/latex": [ "$\\displaystyle - \\frac{3}{4 C_{1} R_{1}} - \\frac{\\sqrt{17} \\sqrt{C_{1}^{4} R_{1}^{2}}}{4 C_{1}^{3} R_{1}^{2}}$" ], "text/plain": [ " _________\n", " ╱ 4 2 \n", " 3 √17⋅╲╱ C₁ ⋅R₁ \n", "- ─────── - ────────────────\n", " 4⋅C₁⋅R₁ 3 2 \n", " 4⋅C₁ ⋅R₁ " ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eigens[0][0].subs(((r2, r1), (c2,c1)))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- \\frac{1}{C_{1} R_{1}} & \\frac{1}{C_{1} R_{1}}\\\\\\frac{1}{C_{1} R_{1}} & - \\frac{1}{2 C_{1} R_{1}}\\end{matrix}\\right]$" ], "text/plain": [ "⎡ -1 1 ⎤\n", "⎢───── ───── ⎥\n", "⎢C₁⋅R₁ C₁⋅R₁ ⎥\n", "⎢ ⎥\n", "⎢ 1 -1 ⎥\n", "⎢───── ───────⎥\n", "⎣C₁⋅R₁ 2⋅C₁⋅R₁⎦" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m1 = Matrix(((-1/(r1*c1), 1/(r1*c1)), (1/(r1*c1), -1/(r1*(c1+c1))))); m1" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "eigens1 = m1.eigenvects()" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [], "source": [ "eigens1 = m1.eigenvects()" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "scrolled": true }, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[ \\left( \\frac{- \\frac{3}{4} + \\frac{\\sqrt{17}}{4}}{C_{1} R_{1}}, \\ 1, \\ \\left[ \\left[\\begin{matrix}- \\frac{1}{C_{1} R_{1} \\left(- \\frac{1}{C_{1} R_{1}} - \\frac{- \\frac{3}{4} + \\frac{\\sqrt{17}}{4}}{C_{1} R_{1}}\\right)}\\\\1\\end{matrix}\\right]\\right]\\right), \\ \\left( \\frac{- \\frac{\\sqrt{17}}{4} - \\frac{3}{4}}{C_{1} R_{1}}, \\ 1, \\ \\left[ \\left[\\begin{matrix}- \\frac{1}{C_{1} R_{1} \\left(- \\frac{1}{C_{1} R_{1}} - \\frac{- \\frac{\\sqrt{17}}{4} - \\frac{3}{4}}{C_{1} R_{1}}\\right)}\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$" ], "text/plain": [ "⎡⎛ ⎡⎡ -1 ⎤⎤⎞ ⎛ ⎡⎡ \n", "⎢⎜ 3 √17 ⎢⎢───────────────────────────⎥⎥⎟ ⎜ √17 3 ⎢⎢───────────\n", "⎢⎜- ─ + ─── ⎢⎢ ⎛ 3 √17⎞⎥⎥⎟ ⎜- ─── - ─ ⎢⎢ ⎛ \n", "⎢⎜ 4 4 ⎢⎢ ⎜ - ─ + ───⎟⎥⎥⎟ ⎜ 4 4 ⎢⎢ ⎜ \n", "⎢⎜─────────, 1, ⎢⎢ ⎜ 1 4 4 ⎟⎥⎥⎟, ⎜─────────, 1, ⎢⎢ ⎜ \n", "⎢⎜ C₁⋅R₁ ⎢⎢C₁⋅R₁⋅⎜- ───── - ─────────⎟⎥⎥⎟ ⎜ C₁⋅R₁ ⎢⎢C₁⋅R₁⋅⎜- ──\n", "⎢⎜ ⎢⎢ ⎝ C₁⋅R₁ C₁⋅R₁ ⎠⎥⎥⎟ ⎜ ⎢⎢ ⎝ C₁\n", "⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ \n", "⎣⎝ ⎣⎣ 1 ⎦⎦⎠ ⎝ ⎣⎣ \n", "\n", " -1 ⎤⎤⎞⎤\n", "────────────────⎥⎥⎟⎥\n", " √17 3⎞⎥⎥⎟⎥\n", " - ─── - ─⎟⎥⎥⎟⎥\n", "1 4 4⎟⎥⎥⎟⎥\n", "─── - ─────────⎟⎥⎥⎟⎥\n", "⋅R₁ C₁⋅R₁ ⎠⎥⎥⎟⎥\n", " ⎥⎥⎟⎥\n", " 1 ⎦⎦⎠⎦" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eigens1" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle - \\frac{1}{C_{1} R_{1} \\left(- \\frac{1}{C_{1} R_{1}} - \\frac{- \\frac{3}{4} + \\frac{\\sqrt{17}}{4}}{C_{1} R_{1}}\\right)}$" ], "text/plain": [ " -1 \n", "───────────────────────────\n", " ⎛ 3 √17⎞\n", " ⎜ - ─ + ───⎟\n", " ⎜ 1 4 4 ⎟\n", "C₁⋅R₁⋅⎜- ───── - ─────────⎟\n", " ⎝ C₁⋅R₁ C₁⋅R₁ ⎠" ] }, "execution_count": 23, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eigens1[0][2][0][0]" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle - \\frac{1}{C_{1} R_{1} \\left(- \\frac{1}{C_{1} R_{1}} - \\frac{- \\frac{\\sqrt{17}}{4} - \\frac{3}{4}}{C_{1} R_{1}}\\right)}$" ], "text/plain": [ " -1 \n", "───────────────────────────\n", " ⎛ √17 3⎞\n", " ⎜ - ─── - ─⎟\n", " ⎜ 1 4 4⎟\n", "C₁⋅R₁⋅⎜- ───── - ─────────⎟\n", " ⎝ C₁⋅R₁ C₁⋅R₁ ⎠" ] }, "execution_count": 25, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eigens1[1][2][0][0]" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle - \\frac{1}{4} + \\frac{\\sqrt{17}}{4}$" ], "text/plain": [ " 1 √17\n", "- ─ + ───\n", " 4 4 " ] }, "execution_count": 22, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(eigens1[0][2][0][0])" ] }, { "cell_type": "code", "execution_count": 26, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle - \\frac{\\sqrt{17}}{4} - \\frac{1}{4}$" ], "text/plain": [ " √17 1\n", "- ─── - ─\n", " 4 4" ] }, "execution_count": 26, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(eigens1[1][2][0][0])" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}- \\frac{1}{C R_{1}} & \\frac{1}{C R_{1}}\\\\- \\frac{1}{C R_{2}} & \\frac{1}{C R_{2}}\\end{matrix}\\right]$" ], "text/plain": [ "⎡-1 1 ⎤\n", "⎢──── ────⎥\n", "⎢C⋅R₁ C⋅R₁⎥\n", "⎢ ⎥\n", "⎢-1 1 ⎥\n", "⎢──── ────⎥\n", "⎣C⋅R₂ C⋅R₂⎦" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m2 = Matrix(((-1/(r1*c), 1/(r1*c)),(-1/(r2*c), 1/(r2*c)))); m2" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[ \\left( 0, \\ 1, \\ \\left[ \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]\\right]\\right), \\ \\left( \\frac{R_{1} - R_{2}}{C R_{1} R_{2}}, \\ 1, \\ \\left[ \\left[\\begin{matrix}- \\frac{1}{C R_{1} \\left(- \\frac{1}{C R_{1}} - \\frac{R_{1} - R_{2}}{C R_{1} R_{2}}\\right)}\\\\1\\end{matrix}\\right]\\right]\\right)\\right]$" ], "text/plain": [ "⎡ ⎛ ⎡⎡ -1 ⎤⎤⎞⎤\n", "⎢ ⎜ ⎢⎢───────────────────────⎥⎥⎟⎥\n", "⎢⎛ ⎡⎡1⎤⎤⎞ ⎜R₁ - R₂ ⎢⎢ ⎛ 1 R₁ - R₂⎞⎥⎥⎟⎥\n", "⎢⎜0, 1, ⎢⎢ ⎥⎥⎟, ⎜───────, 1, ⎢⎢C⋅R₁⋅⎜- ──── - ───────⎟⎥⎥⎟⎥\n", "⎢⎝ ⎣⎣1⎦⎦⎠ ⎜C⋅R₁⋅R₂ ⎢⎢ ⎝ C⋅R₁ C⋅R₁⋅R₂⎠⎥⎥⎟⎥\n", "⎢ ⎜ ⎢⎢ ⎥⎥⎟⎥\n", "⎣ ⎝ ⎣⎣ 1 ⎦⎦⎠⎦" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "eigens2 = m2.eigenvects(); eigens2" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$\\displaystyle \\frac{R_{1} - R_{2}}{C R_{1} R_{2}}$" ], "text/plain": [ "R₁ - R₂\n", "───────\n", "C⋅R₁⋅R₂" ] }, "execution_count": 20, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(eigens2[1][0])" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$\\displaystyle \\left[\\begin{matrix}\\frac{R_{2}}{R_{1}}\\\\1\\end{matrix}\\right]$" ], "text/plain": [ "⎡R₂⎤\n", "⎢──⎥\n", "⎢R₁⎥\n", "⎢ ⎥\n", "⎣1 ⎦" ] }, "execution_count": 19, "metadata": {}, "output_type": "execute_result" } ], "source": [ "simplify(eigens2[1][2][0])" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.8.10" } }, "nbformat": 4, "nbformat_minor": 2 }