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https://github.com/rodneyosodo/qc-mentorship-program.git
synced 2026-06-23 04:10:29 +00:00
🔨Adding mathematical explanation of finding lowest eigenvalue
This commit is contained in:
0x6f736f646f
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Eigenvalues and Eigenvectors"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import matplotlib.pyplot as plt\n",
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"import scipy.linalg as la"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"tags": []
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},
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"source": [
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"Let A be a sqaure matrix. A non-zero vector v is an eigenvector for A with eigenvalue if\n"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"tags": []
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},
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"source": [
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"Rearranging the equation, we see v is a solution of homogenous system of equations"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"tags": []
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},
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"source": [
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"where I is the identity matrix of size n. Eigenvalues of A are roots of the characteristic polynomial"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"tags": []
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},
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"source": [
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"The function `scipy.linalg.eig` computes eigenvalues and eigenvectors of a square matrix "
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 1 0]\n",
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" [ 0 -1]]\n"
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]
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}
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],
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"source": [
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"# Let's consider a simple example\n",
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"A = np.array([[1,0],[0,-1]])\n",
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"print(A)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"# The function la.eig returns eigenvalues and eigenvectors as tuples\n",
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"eigvals, eigvecs = la.eig(A)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[ 1.+0.j -1.+0.j]\n"
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]
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}
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],
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"source": [
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"# The eigenvalues of A are\n",
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"print(eigvals)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[1. 0.]\n",
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" [0. 1.]]\n"
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]
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}
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],
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"source": [
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"# The eigenvectors of A are\n",
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"print(eigvecs)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[ 1. -1.]\n"
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]
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}
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],
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"source": [
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"# If we know the eigenvalues are real numbers, A is symmetrics, then we use the method `.real` to convert into real numbers\n",
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"print(eigvals.real)"
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]
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},
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{
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"attachments": {
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"image.png": {
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"image/png": "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"
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}
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},
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Symmetric Matrices\n",
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"Symmetric matrix is a square matrix that is qual to its taranspose\n",
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"\n",
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"\n",
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"\n",
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"- The entries of a symmetric matrix are symmetrics with respect to the main diagonal\n",
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"- eigenvalues are real\n",
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"- eigenvectors are always orthogonal"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 1 7 3]\n",
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" [ 7 4 -5]\n",
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" [ 3 -5 6]]\n"
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]
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}
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],
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"source": [
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"A = np.array([[1, 7, 3],\n",
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" [7, 4, -5],\n",
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" [3, -5, 6]])\n",
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"print(A)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 1 7 3]\n",
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" [ 7 4 -5]\n",
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" [ 3 -5 6]]\n"
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]
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}
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],
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"source": [
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"# Transpose of A\n",
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"print(A.T)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {
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"tags": []
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},
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"outputs": [],
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"source": [
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"eigvals, eigvecs = la.eig(A)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[-7.00792852+0.j 7.03594585+0.j 10.97198268+0.j]\n"
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]
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}
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],
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"source": [
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"# Eigenvalues have zero imaginary part\n",
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"print(eigvals)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"[[ 0.68415383 -0.62878419 -0.36954564]\n",
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" [-0.61391034 -0.22292725 -0.75724338]\n",
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" [-0.39376087 -0.74493885 0.53853365]]\n"
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]
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}
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],
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"source": [
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"print(eigvecs)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {
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"tags": []
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},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"0.0\n"
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]
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}
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],
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"source": [
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"# Check that eigenvectors are orthogonal\n",
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"# The dot product of eigenvectors and is zero \n",
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"v1 = eigvecs[:,0] # First column is the first eigenvector\n",
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"v2 = eigvecs[:,1] # Second column is the second eigenvector\n",
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"print(v1 @ v2)\n"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "PyCharm (Qiskit)",
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"language": "python",
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"name": "pycharm-f8797ed6"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.9"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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