lectures/lecture6/lecture-6.ipynb

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    "# Lecture 6: Eigenvalues and eigenvectors"
   ]
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    "## Recap of the previous lecture\n",
    "- Linear systems\n",
    "- Gaussian elimination\n",
    "- LU decomposition\n",
    "- Condition number as a measure of forward stability of the problem"
   ]
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    "## Today lecture\n",
    "Today we will talk about:\n",
    "- Eigenvectors and their applications (PageRank)\n",
    "- Gershgorin circles\n",
    "- Computing eigenvectors using power method\n",
    "- Schur theorem\n",
    "- Normal matrices"
   ]
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    "## What is an eigenvector?\n",
    "\n",
    "- **Definition.** A vector $x \\ne 0$ is called an **eigenvector** of a square matrix $A$ if there exists a number $\\lambda$ such that  \n",
    "\n",
    "$$ Ax = \\lambda x. $$\n",
    "\n",
    "- The number $\\lambda$ is called an **eigenvalue**.  The name **eigenpair** is also used.\n",
    "\n",
    "- Since $A - \\lambda I$ should have a non-trivial kernel, eigenvalues are the roots of the characteristic polynomial\n",
    "\n",
    "$$ \\det (A - \\lambda I) = 0.$$"
   ]
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    "## Eigendecomposition\n",
    "If matrix $A$ of size $n\\times n$ has $n$ eigenvectors $s_i$, $i=1,\\dots,n$:\n",
    "\n",
    "$$ As_i = \\lambda_i s_i, $$\n",
    "\n",
    "then this can be written as\n",
    "\n",
    "$$ A S = S \\Lambda, \\quad\\text{where}\\quad S=(s_1,\\dots,s_n), \\quad \\Lambda = \\text{diag}(\\lambda_1, \\dots, \\lambda_n), $$\n",
    "\n",
    "or equivalently\n",
    "\n",
    "$$ A = S\\Lambda S^{-1}. $$\n",
    "\n",
    "- This is called **eigendecomposition** of a matrix. Matrices that can be represented by their eigendecomposition are called **diagonalizable**."
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    "### Existence\n",
    "\n",
    "- What classes of matrices are diagonalizable?\n",
    "\n",
    "- Simple example can be matrices with all different eigenvalues.\n",
    "- More generally, matrix is diagonalizable iff **algebraic multiplicity** of each eigenvalue (mutiplicity of eigenvalue in the characteristic polynomial) is equal to its **geometric multiplicity** (dimension of eigensubspace). \n",
    "\n",
    "- For our purposes the most important class of diagonalizable matrices is the class of **normal matrices**: \n",
    "\n",
    "$$AA^* = A^* A.$$\n",
    "\n",
    "- You will learn how to prove that normal matrices are diagonalizable after a few slides (Schur decomposition topic)."
   ]
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   "source": [
    "#### Example\n",
    "\n",
    "* You can simply check that, e.g. matrix \n",
    "\n",
    "$$A = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$$ \n",
    "\n",
    "has one eigenvalue $1$ of multiplicity $2$ (since its characteristic polynomial is $p(\\lambda)=(1-\\lambda)^2$), but only one eigenvector $\\begin{pmatrix} c \\\\ 0  \\end{pmatrix}$ and hence the matrix is not diagonalizable."
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    "## Why eigenvectors and eigenvalues are important?\n",
    "\n",
    "- Eigenvectors are both important auxiliary tools and also play important role in applications.  \n",
    "\n",
    "Can you give some examples?\n"
   ]
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   "source": [
    "## Applications of eigenvalues/eigenvectors\n",
    "\n",
    "- **Communication theory:** theoretical limit on the amount of information transferred\n",
    "- **Designing bridges** (mechanical engineering)\n",
    "- **Designing** hifi-audio systems\n",
    "- **Quantum chemistry:** all our microworld is governed by the **Schrodinger equation** which is an eigenvalue problem:\n",
    "$$ H \\psi = E \\psi, $$\n",
    "- Model order reduction of complex systems\n",
    "- Graph analysis (PageRank, graph clustering)\n",
    "\n"
   ]
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    "## Eigenvalues are vibrational frequencies\n",
    "\n",
    "A typical computation of eigenvectors / eigenvectors is for studying \n",
    "\n",
    "- Vibrational computations of mechanical structures\n",
    "- Model order reduction of complex systems"
   ]
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/ACEH7TXzV3xuZ3k/EIslS3mcjA/yEO6POvszWqxOjqW6OYoqSG2RmJ95sJvX3XC297feDPXHbC25yLeF4D8czeYed/QQrxe+AgdfbzB1m4ous/8AeL3wcXXWQjrCo3wb4CzkqW8w+lT6xS74fSYhC5Sp9PMW8D8JlYH9JbUjt0cJbQy9TxsI6z6xX1czXtcM9ZIF4lSHXCmSpF74Iutus3CEmK9Jhd4cWF4i8ZvC8O3TKB/STa/kKmB26eEXV1LdAcMyOMSOMyOM3lu3TLwvId4Xguk3j1I/EQwge3Egu01fyFLxE6ep7Mrt6pai77RxJiLeATdN3xYZS/GUqP6S3ynha4L0N5nNbIykbfCMyftGGLw4fd2CN26RuMcISvI+UrJJrcSUk3AZ+es4mC7rB6kIJrsSs6yLpKniUM17V8OBVyOP5lN6FK+RwtNpH4lNTnVvVlZTEwTcSmwzp7aGNvOWM5f6SZlPFiVnNiWeU/KEK/USWuxWQPxKW+peHEooJu0UsaOe1cFF0jkav4bfdi6ByBMkECJ3tFpxYAMzv6Swne1FKyfhxJM79mpCBB8jCN96RFA/MQo5m3jKWtYJ8oh39Izf6QR+0tCC7weYRX2pJaMJxECVSPeLfhEUiD1RzFrc2656w9BWesVFg/HcBddZHkmKaMeSYhC5SYXeVMEw/vgtc0k3EaCCptUxNJU8SnQMihVo7mAyuZXdNxX740GpbVczNX8odue6yCVJXyOEbhmtkqSVBUlMkw9G4W0aTL0BerFKkxOgctCzpPlHqt/RyiKAsK+Hs2a0NmVne3EZeYbqpvSykCeYJon9ZHEqSiepDrLEM169SEFZh0mf64Igma5SxvHRt2UUlZbibCNLqZfYMNX94wWmQVNxYxuZ+B/SWCTECwv9JbUD3cKmZepLDS1ZdNxDQW1ehXlhqh90lxmfhL0hC5jESTWkPrJXV9YGkGnjdmia0ys83pNbp7ipZGYxNe/E3vAo9viTQTcRU74lUE3EoQ0eZv6FKh6ktM27lWMrJNbiFraCp4lNpqWsupIV8hzmhqbpVU22ofmiloVMFYpPyavVmVTEyVPpLnSE3bAW+rJl3hS0DrvFH9ZvbMxS5ZNbiQht8y8IZaQqt7Y1ZiLQZkt5YsM6msxJuSzXYFNqSZeEXpqpuwZQtYVc3FaPVXFGVle/aFhVv2RAYybikYhwd+2BJ0090zL92VkD3Vlv3hYm9Ztl6B5HVaq1ilzZ7KyNfvCVmT2V1pYm51xVca+eMeg79l8O7I2bP+l07fdk2kTF6q0gTIECv4ZY1LegySVbbl4X7tx6fJHd7reQDhkY/eQd8G+IQmgQkmH98QJUbheMXjkYEmk5lOm5F3SnL6R+I6Zpp+zUtsptWc5lX+U1WsOZjHu/pCAAAQguMfjcYC8tCVviwoH9BS3kqkqbS1tbQP6S5zJ+wYydJWGmndmpvww2c8nf0sQp3Hq7hkYhyOEPBACAgC0f0jIuP5VIHWY+KkX3Zhsz+Vjc5NxUv4Zic2h7xjCb6/i+LqhRvaOb4r77RF4g+f3/AImmPMWmmu+Uooy/0snaqbsGh12/ZKYW9lxNtr5OyjMGBYdZ9BFkmuEXgCDe6XmupWUxGZ943vDYaMdWp2MfmzrvJPeFtka/0CIJu0UbkcRBzKEN1qF7aaNjEVbmtz17qZfdmSntICMpdt9H7w6VqX5opzikTjVjoFfNdQr7AGDkLnR92/W0ovEXuj3tnUOzPa3T9JkM5T94prddp2t33ZlU7xQzbfOuGmUytJ0tIppM6e6lX3ZS0CdlG/3k0ZYk674N3d9mGiOORlX7Mf2m8SUvsDezlLZm9wA/mz2yFnJNdBG3nKnVDsrsWFluXx4+QLGluGrb3ZUX2ZnIq/bj+l69ZatVXxxnlfR3ZnCsa8TydoGJGr3szSP3gjWc26r6dl8ZN1LlU9Rmi7tHt7y/dhqzJ9/Uwtf3JKzmcv8ApUPqHScmSmbL2W5OtbyHd3nOa+julV7t4W2WI7SxszcKGq3TZ6yJsKe3/DRkJ6xelm85nHr7Q6/6eY7R6Hz/AHi7yKOGDznox6MjXj0bjMTfCIvIrzCLxmhZwTelTpWk5uzU5LH8qnUNHpdCoWjazf0mPd/SbDW/hMRIFnLwvGACD949eRYxDv6QhKvF77ykK+4sKSEsSssma5Tocj3QL7sx+W0C9KsaZ37K0Lc8r+ZiFITa/mYigMgLEFoIFwfKCIW0dA9tyq8nsEDoek+Kl/DMtqWpXptX8Q1ul6OfqPCjx3+czFXonMpZWa1LX+8Mc32PE8qkGMkCM6HBs0dsFZqymj+0QsINnuXp31dv/YjNoQfL7zvNzaBy209NbIbdMkyWmwbhmnYYjmy2Llg/jLYM/rPMmltTyGWsbyv/AAHTJM1pV4lghuGX1Ci8SwQ3Bbn6Uc/2E38BNpMnnf6qb8s1r6kfwqifhjfw8/Rc7cIObP6lfMMqpY1hRP0koMiy2qrMGbxGc2g7QZaqdURmeOByvyzX0sEly8CuHqzdMg0lVPjyoWHxPeLG7ewyD+lpnrIY6nrUMazcchPvTp4cd564YzQ6/J3eKxm4jP8AxVqenhNne3QIjr2TwBbMQabqUxXG3eewaeooJWpmVUd23fILfNanwqkYx1yu+VZ0JzcYz4Eq2xuWByw01ls61SrJFNHZ92XvX8wX68IMyrPrJ3kOOka3pm3i8JlZIVTiNbPUyvhxNvCokT08pCDMbs8PQvKFJDbRwr9+WEDt0MticfjHKR2it4S3DGv6ZmipVH9Aw9rIWc+ZRT4KskXIN0lZTRNdGm7DQjOaZXlZbXH4KB3pFTu23fjGKjOGbHhsIz1jy42ySvaF5l5TkKZdIs879p5ELN84VZL4YKa7zlZYrcK8YRw4/wCk2eWcCKvMqmVuJt4ohEduZielMviIdXmUFNh0r25l0My4IWXiuckx1iqUU+dyz8KQbhXH4MqtwuZ94U3hAur1Da3CryKRY9Qp9ZvoCdHRp4VJMeTo/Cyk3Xm5KAE2CmZuVXkOPKr+Ufjckpk9SzW7ib8stKTTFV9hNH+GFqiNLiT1Y0MGlazhtgf8stqDROaz9zS/xyFoYVIWXE6JpCZtypMpNkuay96kNCvndzTZFoylo41SbMEuTvCFsbrObszGRpcdjznT2Tvj21VNVWfYkL4NyVOVamT2zuY5ZJTWiEpm6TrPVsq8NNvBd+VJ/gUGaHL+pt0DPwbO+PCjyexGdafUlNFww0dBB+GMPq2Ve7SjT2Izg5rBklSvNTVP5Zb0GSVT4XR003GaZ9VVX/af+WQp9T1Lc1S4DmTaSzeV7dwkC+d5C8n0fWLgytPR/wCuYzMmd1Lf4mb+MjT1LvxNi8jFoXL6AibHpnzKjjb1A+J+XdPzz/8AQzm+9PKNvNL4UcLaf4q5PFjdJLNP7AbnIIP8HNVe3IZbfP4lmQbe9uVX/LA2cGd5Qlu7yxI7O7LGfWCstsCQwL6iHO46aqbHhpql/wAMsIMkzJuJaV7fXA0D6tqlwtVypqtQ1TcW9HoNJZg+HEiJ+ILk0rZw1NVDSsEKJ84nbmcR8KSmk6hktNh+k1289iMYSpyFfm0VTXeuBRJXu+JK7V8OEsJ8yi6Laaj3ftjPWZ25VsYLQky2pfHhXdr53JXwDL0XNLDGvvA7d+/n4QpKZGxbieewNOcCOpqneOVGrM1Wjo6h7b703cZo6SjaXBmjTkMztFplnpJILkjYNKPn98bsekRgSZ0txtI3y4kPa7hskr1qKbdyLxR8jm8gpvKcM0ZmUtH2arxObOCvqpcPGXN5M3Q0miXHtHQjT5rTJzOYbqcr8zOefA/mYx0aUa2TUNMvi3hCq9WwLgqxxcXnKOTKlXDmGOporcryewNCjQSaqV8OGLdkJ9T4+FQgy2Vu7ppiwgyGpb/DfxjRdFS+p5Wx4VIsmp5/Khqk03U9HdQjD6Yn+yT8wz0KMymrZ+nukH/ja/iiLz4sT/ZII+LcvkQ7oUV8esF6OJHJUGqqZu8Rxfxen8MSEV8qdeamOaCVHqqh+yce+MlD0cjkaPKn+yJPxefo5UJu6XBqSm5Y1eRj2fU7twxwf6x6DTzeKyMlJk6rzdoXCbioepqqnHidIF9QXHlqrzF11NV5cBfViyiFBTN0cJZ0lGzYceJNghVcBd6qAylGi4jNWPdZUhTvdiHU344Rf/baH+AeTXPotWhoIPs33fdnNoKmV8bVv/LJMeW1zcSwTSGzwOj/ABtnb7H8siyatrF5Zf8AlmSpMhzJ+WJ4yxg0lmDczbsC2fWFd9uIfWFY317kWk0NK2PFUwx+2WEez1V7zMKZPxAhWSZ9K3eSuM/CV395okyHJ6bv8whkb1O1ET02QdHzz/lgZzrLty4uIepnbwuW075Gn+Kmk9glfGrKIMOGKpqgM5+kt4XFpR1TeCY0ce0KmXHsMv4fXCfX9S3d0cIWpo8hrH8LklNGVzeEfk1hmb8KwQoMSV+cS4d+Q0WdJs9qm7ySGP8AEH59nVlrSVKfxmc6nmT49tUv+YMVeSM2PFUzSAo1Xxby+LvKni94N9ZymDh3+8b3ZnaDKolx715PxB50pExtbtGCKLCTOMvXHs4nk/0AmrYE5aN5CvevposOGK8eyaaWs3jR024jSPnC6LCPWcr9xl8Ke2MvqrOPqYKOMrJ6+dWsWJ5PXLDMsqrkxXculu7C8w+odRv9akK+oQqubOp8LZ8we1+csJ0nWBbn4vrBnJoVVZut1L8fdoDNSx0FSuPDXVg91Cl6O3nmnm94Tcpelgdmnn36+Qr581y1KlpY+0sBmlRw0a8O6cHtXHhi3alXX6tibiWK8r59Ws3h3ZGi6NPJ0o0fJa4ud4lkVpHQ55PnEr4s1zh2svicaFG2jzWhppZGZnmV/AQvjbFxbuBI/YMzHlrNzMSoKNF5lcjRpQ9V6knbhj7P2Cvneep5ry6SFfq0JkeQzy/37kaDiGtsv6vKuHmKag72P2zp+27Sq0K0st++kePtDm+TfOYf2byPpL5tHTsiyqWWp3sdG931bvGbqk09WPwyWQFnBnzrHGq8u7ET5q7cIQippJlx7SqJUem6RMelp+L3gzxy+Fwej8xiJllHFw2pOCV6L3cEP8BGghRe87Qe4Pq1csTfhufw2Ri/hWduZysEHOgto6/HxCOuFfGLHMSuuMN74QFho4f3wb4bsBE9JCT4ABYXeAgReQssQ7qo28xGd2LcSZKn0kaep9IzINhJ/fCLxsA4fj2nRLy5en5YPtLqm7mjSNfdlRvsvTlZBEmfUqcKls81hJrnNXx5Ug9iMZ+Eszn4t+8bEL4yI3Kjlg9ZKkcb7p7Zu7BmhT5VmEuN0lU/5gj4tytzVbi/hWufu4kj9st4KOuegaqvSPtN3YDNBoNMIvM4/wDAMC+JCpfLapuZ3LrSGm4p4qzrbzb76vtATgb+CqWLikYcpJstV1Wx5Dyr0rB0cUrxqnncsNJzZfQwV1NO6dt2kDhFEar1Dl8GNscE35ZJynNesRSSwQWbmPx/WFY+ZZe3EwQatpIMGWNechtQufOKxntshRd2XXVql6SnnjXdtNHvO8MrV65TotjgT/WVlXrysltW5N3B3aIHWtoKCqWWNpHd495x3yFhqHJ0e7dy2Kn/ADDmU+qqp/rXjIUmcTt43kA6bkVNBR4yPMyTK/gK/MqyhR7jnj1kr+IRGkrhdHSavWyMq4WpwFTV62dcGWNecyqZVKSo8hduYaCzk1nP4SFPq2sf69/zAj09bhxMP/A8C8zmejqr+GKp/r5hHWZ34bncuv1fF53CPMqZu4V3Yi4pUo528I8mWu2PEWiTVUuPCjoWdJk9c/eNu1Oimjy1E4ndB6BKRe8V5DTUmlUXikd5GLCPLaWLmUDJ8D9zATaTJ55fDuC/nrIF5SNJnADFJpuzvnJvU6aLwlfPmVxXyTM2JmLrrkUXKo98Ks2FqlLSIz48RpMtpok4pDbmMLtYppXy1nbtO0OOdW3VVEnrn1NrPNaP4JqoGTn7t/vj5Up3brEKv8sL2Ymi30xlOQvuo2kXd3x+MuaCjposO1W8qYM4vVWj5d2M77FsQxaOfMoE4YkQoquZWxuGb2AOABYgAjABYAAsXYQED43uR9EALAF3kV5iA+IeYYkcZkNlpLzDN4yAD0jjMgAEmxAsQHAAAAzoHT0FLWVT1rpBT7jdpDu972w3Pk9MmN29vOe/GSp+1chVeaytjxO5A6tAmXphxMg/X6kod3Gl/DD3ZxiSslbxOHWXDR0qv1bAuNsa7wiz7QnWPdRLu4/Ic83jhY7YluNTV6wnflbdqRn1PU+GV4ylgoJX5UcsINPTt4HISHzuV8OJ3Gess3iLZNNsvMTYNPJ5g4z8bs2JKsdsPGamPKoEw5kFvU0sHiQ5o6y0dA7f3ExMklbwl78PLywQI/r7sZ65mc+NsMBF1maTTbNzNuyT8A00XFJKEeVZrLj0SLu19Qt4NDSvh2krxkaCrepo14V7T8MPhJOi2Gjvbzm3ynR9HFhxJvGLPc0kHKiRl3W5ZJU1j42xQOTYMhzOfDm3d/3Z0b4YpY/DxEWfU6csaC6GGj0TVtjxSuSoNnV2PbYuaZ9SMxFq84Z/E5AjUmiaSm4rf+YTOpwRcqoQJKl25bxux2xAs0qYl8IifNVXDhGYMnnbwlh8W/R2rIhoKKfNWbxEKesZvEaf4EoV4pJd56gbnL05YHkDjETu7Y8ri0pqmXhWJzYvUxN3absY6yxaVZBp6VcFadkT1N4D0ypy9oWe+uEWKFXV6O68oOjsWdigBRV+VNUoySNwlNT6JpopVezkNtGLdAlDgprcB7cixYQQkI9YEY4W4YsF7klIguwhcELcjnVvQTI7QeZFwC6Idgu9VGJHbwiETzB0/eDzMwAADA+MBwiQZFyCDZIAADgESCxEhAbvC8RIIAXeF4gepKN5XVVV5LyGjlnwDL5XJUGm8W5iT8cIvCryDPxkll7uBzDQTKTTCNzM5Oj0xEv95XI+ZS42rFu/xCxg03ms/M+4UaA+CqNMe0cXI+Wp40LCk2bu/FNO7lnBs3pvGqDQZV9Q0sHK28b1IwfWDNwwxPIdHy3RlDF9QkhbfA9HFhcsSRhDj6TZlVY9nFu7/OTYNJZnPxNLu/YOoPNTJ4UjIs+dxJhwgY+DZo7cU0s0hdZZs3pV4mXeMLn1sy42rE7jM+qp3w5d2FtbluSU1KtqqgVdfTRGFkzWV/GReOXHzmg20moYF5SBV6ku7tSopMqdvCWEeTt5TOgjPnErEWesdvEaBNNs3iSMeTIYE72dIy6DLIjSEpKNm8Joo/g+D+/f+xGIkz5E7iJC4Qcur6TTcsv3ZNfSqxYXS1UMHqOV9fms8vjeP1EK/jfHtG3hFBdX5fFw379vYHPhuJO4g3nrlDuRxENM0raTO53w4ew9gr56l2x4mECLA4XeA2ADgALLQBYgWAAAAAC0QfSmIaGAjJqUdw+lBbzHNFURaeG4kx0Y5vkThUi1dY7co0aJPCmHEQ56m7HhIvpbmAkPvMzDYi8ReBKG5HGbxDzF5oPXiN8RrzwCVvhG+EDZYcAALcAAMPMQk5eNyORd96REj3BR+8lUFM748K7wmaX0xPX48KvadapMkockjWWXGHebvxi6XP8AJdDVdTjdJhuFNHvsuyWNlZd/UFFrvaFLUYtFRNuF86GCvd+KRnnbzvIHF7SaVpoMOFIf4CVHlsSeFDMz6nchfDcr4nibtzAkC+FCV1mJMOY55JmTNgRXqXblLHQJ9QxJylfJqryqY9IXbwuSoMtqn7uB5CVrOr1JO3L2ZFfNXbDnLfLNE1T4XTMkHtlhHoyhi+c1X8EhTC7HvmT+YbTevj8jm9gy3I4uK95h581y+DuIEkLoXYmCmlbwOW9Jpiql8G79suZ9T/ZwQoVk+azy94woXPR6eSmxvq5YY19SQfTPssgw7FetN7spdzdxMG59JvmhNn1nOmPYwIMvqTNZ8O9SBfdhZh5QsAhb6ubvql5B6NG8Q/YFgAiC7AFlhFgWCwICLBdgAQ6AAWHDLoFgt+EjPMA9YIIvWWblGKuaWJL7N4c0XRaRkmOmZjLZFqS+WyaCaNfObmDNaZUtjR5G9ci5RGSjZhcdGvT2jbse64zfdjLpfjxHWmZ6yKLxJID1K+FRnqaqHCoBJWP4RnjbmYXepFepVSA+iBIQuuEWepxKoJsk1oy8xCke4WaM7n98F42FhYcvGxYBJAsALUcGxF4SOQk5eMPMMPMNcwWdeYRZi3KTspyeeqkVI8HOpaP2aW4b2pYDmeU6YrKplWGLeHRtPbN1psVlrWhtNHqHUlDki7qBb5t3yHLNUatqcxxa5nRfIhDje51rajoEsoESRvOhyzVGfVNfKzzO/sFdYwJCM0kQQk1IVCNLcAke0tbEo7PyqWdBlU78qmg+Hm8MEP5ZFnzKd/Fu/YM6QQINPS/WOiFpBTUcXeY7wpeIERjuY03w3SwdzFvGGZ9VVLd32ZS2C7Bmq5+fMqmXvJXci8TY8Q4LRAgALCwAjHhEYssOAAFoAsQLAAAWQAAAhYAlJRu3hJkGVP4rDmjRUi0hZi6jypF4pHG33SY8BF1oCUbt4SSmVP0XN2Y91+3DhIc9S74/KdM0mymXmdLkIzzRdPChFkhVuIEQ5dnQt+LHhH+rYNhxDF9o51k60LgoIk5VQesQhPUkbrLGdBZu6hvsCseZiLvsTfMW71KqQp6whcXMIfiGaC3mZhF4WBYW4LwCwLAkAABw8nyDkYxGPAAh3wURO5FAlb4YeYZsF7kOi8RePJRs2PCaTT+j6mst3aPb592NFs3BCz48p0TRGgManDezrwmz0voaDLsL5GR29clZlq2CmuSPFLk8hno4mfBuX5TFeypHYYnVOv3e6Kkbdr5yi1nqqWs4enhMnedSXVuzs2MjbxnGLByMWHCLBD2qKI0/EWG3qQvuEJCLsExGRB5EEIg9GQgWDlgD0ZYRYFgsAsIg5YeDl4QRYFgvhDhIDY4AFrAuMQAALCNGbHhJtBlsr42qphd1CRMWJMFG7+E1WWZIiYXTD0+a0dNy2SMRoqimpNPO3N2ZP6hBBh0yMhWZlqSV+GNbCpd3fG5mNGkIL2rzhV4YVIUlfKxCRBwzWevuAQLvAHQQLkcZNADYSOIkchBEggRI9wi89KD14wOXjYcAgLxF4SWAzeLvAWAgA4WIsCNwkcAAReF4CxZFvLGky2ee3do8l4EKQLLjZ5Fs9qarHtOwU28GjMtyyO+d0mZPOLjlGU6eqanG2OJ/yzbZbsxfoV55UgX6ws6raRS02FtEm8MTnWs6qsxa592r+Qhbexw5LlS8m/kTxlZm20tUxtpE3anMnvbG4XYErfMtSVNTizNK/H4N4Vd7eYRYOWF0ceHqILsC8AsEXjbzEWRyw+8wIgxGToyA3GLEBeBGFxiBcbkILsALwvLDx7YN3hvjPQACHmGd9gNGiTeIvIW+9I9Twu/Kdcof3wu+7Em0mTu1pc0mTxRcUnGF0U0FM78qltQZJf3nZlpHNFBhwkKvzK7lxIaZrBKampsOEQ+cbrDszNvMzeIRI5AsK+vll5nd1K/hEXheUFi4xkLzoeAZvDfAPyTWje+GZHESOWHt8G+IV4XjMPyOMXjMjiAzmm3jcjjN4FoLvCRxAAACBYAAAWAAAgIPL2H4IWfG1TQZLo+epx5d2ouMyiYsXOU6bqanFVjU6BkuhooGvkZJC6r87paDDwXJ4EAotPbN06Lqlt2arfUeUw2rZcc41Dryedm3XZqZarr3n5sd4Buc92kSNclMu7XzmGr8ynnx7SR5CLYLsICEQXYLAtAAcAARBdgiMevCyJHtIU8wudyMAALAILRBYRgACJAvEXhaLeF5C3wb4gWF4PMV++xDmOaOpu+EdZGdyLgpmbwkXBez8pJgoHbHiLTKcnu4mU0cFAqEaNVLQZD5lNPQZbFFhy8QdZVMCrzbMm6LYyHVjVVKJylRV5rdylTJMzcw2aCw6yzCLyNeF5YeCRxm8ReEF3heMb4N8HEq8ReMb4Rvi0ngIzzDHWS1rG8bkci74Q8wcPyON3jN4u8JOAMXi7wHwG4xwKAAASAAObG0OCQQXOWaeqam21Oc2GU6AVcemdiBz+koJZe7R5DVZToaplwuZeE6BSZbR0a8NkdhUZtraCLC2Nt4xzQTcl03R0OF0i8SecRm2sKWluSOyRvUOc5tqepqbrne0or7uY6NNnWsJ5+Vt2pnHmZ+Zt4IF2BZAsXYFhaTkYsIxcaXBwgCTucQ3OIEawXYPWCJAGRiRxyRxuwBkXYSrFGJATMiwvEXhBfKF4xI4h5gH7xiRxnfCLwKy9h6kS4I6N2x5S2oMkdsPIZ6NDMFGTKTKnbHlLTLMhx8WJo4IUiwtZuQia2fyzTeL48Tbs0EGTxQYcTDnwqiYchRZlmrvjavKYi5nrIoivfO18JTPcxFeEugs6vNbivesZhFgWF0WL7gvGLD06gt5hvfCAsLcF7MODdg4WksAAOG5BAuQQ6BoRIIHrAsIcMgSbAsCUawLCTYOWFuIVgWD1guwhaNxC47iUkJNpMqeXlUu6VeSYKZnx4VNbp7Q087LvV3CnRsl0xR0q8qSe2Z6OOX5Lo+efFbl3amzybQdNFjfI28Y09fqGko15k4DC6l1/dcsCkjVV+ZUlAvg4DH6l1yz3LEYmvzKWfG6RnIXEBYV+cSz49o28IQiwXYxeYQLjF2DlhYbsCMcsPAAcGx9ELCCZQUzNiP0FBeWcdMsWFwBSUarhxDNfYh5V19vKxTV9ZcQFu6kaeYhPUkV5gHnm9IdZId4XhCZ1kQ9SRpHEASd9iM71hASBZe+uEXgBYBYgAhvIKZV8I/eqELrno5ivnrPTzHibrZ6y0hz15WPUjG+NBNnqWYiu/pGbwM6OJW+G98MiDfNKTeM3iBsKLkc9G+IXGEvQAC1FiBYBIAADhAALjICBYCwsgWAsJIF2egm0mVSy8qubDItDStxTdmW4wUFG748Kl7kulZ58eXhOp5Zp6mp8LmsjYXX59SUeHMhC2ZybQaL3zGqgoKWhXu0MXnO0i3G2mS8x+banqanG6R3QhDpue6zpqfhXtGMLm2s55cW3buimTnmZ+Zt4Nxl5idV1jy49pi8gwIAsPAIjFxgLAAIWBwbA2YnAACAIhZ5ZR3kWCmZvCW1I6xYBa0ghVMCrzastwtGq/NVXhKKrrLvEFirqWbErp3xYXeIDEiwRYPWBYBGdBBJEWAMiwsAMwAuwQAgBYgtoWIkFjEhAt98N3kO8XeYUbpogYjcXedcLvC8ZvAtJ8bHALUAAAkCxABwAABAAACywEAQAWIRMWLbLchnqMbVUCsjHo6Z3xtVXN7kWzpmxum5TbZZpulpsOUho5fkuj558eLszaZboCBcVabwGgzLPqWgXiZOAxGdbSLlZYF3YcbqCgpKNeFU4CiznWdNTY9nzHMsy1JPPizM5SzzM+PFibZpa3Ums56q5bt2pnJ693x4mK/xD4UBYgWEkAABmBcYC4wsAAgB4BAsBZ6nEN2FnQUbNaEEQUbMWdJlqrxMWEEKpgM19YqYAE7qmBRV9ZcIr6y4rJHuAKua7EYAAHBscABscGAAWAyF4AIAAssBAi8ILAYvAAkcZkcXI5GkAsBYAQssAABYsAAAACwAABYHAAJNjgAGYG5AANj8FM749CqaDItJS1PN2YAZ9Bucp0BElu8beGtpKCClw4VSOwAJSr851VTUyta6cBz/Odfzy3LHZGoAWtia/MpZeZt4RQAM5lgAFoA4AFrNjgAA3eF4AQgC7z0As3eF4AA+nESoKZmxPALFjSUDdPEXkCIigBAr6+v8pTVdSzAAQhXjdh6AWAAAg2AAAAAAMyDIAFlgAAAgAABsALQRIMgBA//2Q==\n",
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"400\"\n",
       "            height=\"300\"\n",
       "            src=\"https://www.youtube.com/embed/VcCcMZo6J6w\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x102d2fdf0>"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import YouTubeVideo \n",
    "YouTubeVideo(\"VcCcMZo6J6w\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Google PageRank\n",
    "- One of the most famous eigenvectors computation is the **Google PageRank**. \n",
    "- It is not actively used by Google nowadays, but it was one of the main features in its early stages. The question is how do we rank webpages, which one is important, and which one is not. \n",
    "- All we know about the web is which page refers to which. PageRank is defined by a recursive definition. \n",
    "- Denote by $p_i$ the **importance** of the $i$-th page. \n",
    "- Then we define this importance as an average value of all importances of all pages that refer to the current page. It gives us a linear system  \n",
    "\n",
    "$$ p_i = \\sum_{j \\in N(i)} \\frac{p_j}{L(j)}, $$\n",
    "\n",
    "where $L(j)$ is the number of outgoing links on the $j$-th page, $N(i)$ are all the neighbours of the $i$-th page. It can be rewritten as  \n",
    "\n",
    "$$ p = G p, \\quad G_{ij} = \\frac{1}{L(j)} $$\n",
    "\n",
    "or as an eigenvalue problem"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "$$\n",
    "   Gp = 1 p,\n",
    "$$\n",
    "\n",
    "i.e. the eigenvalue $1$ is already known. Note that $G$ is **left stochastic**, i.e. its columns sum up to $1$. \n",
    "Check that any left stochastic matrix has maximum eigenvalue equal to $1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Demo\n",
    "- We can compute PageRank using some Python packages. \n",
    "- We will use ```networkx``` package for working with graphs that can be installed using  \n",
    "\n",
    "```conda install networkx```\n",
    "\n",
    "- Other packages to work with graphs in Python are [igraph](https://igraph.org/) and [graph-tool](https://graph-tool.skewed.de/)\n",
    "- They can be useful in your projects"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- We will use a simple example of [Zachary karate club network](https://en.wikipedia.org/wiki/Zachary%27s_karate_club). \n",
    "- This data was manually collected in 1977, and is a classical social network dataset."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import jax.numpy as jnp\n",
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import networkx as nx\n",
    "kn = nx.read_gml('karate.gml')\n",
    "#nx.write_gml(kn, 'karate2.gml')\n",
    "nx.draw_networkx(kn, node_color=\"red\") #Draw the graph"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Now we can actually compute the PageRank using the NetworkX built-in function. \n",
    "- We also plot the size of the nodes larger if its PageRank is larger."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING:absl:No GPU/TPU found, falling back to CPU. (Set TF_CPP_MIN_LOG_LEVEL=0 and rerun for more info.)\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pr = nx.algorithms.link_analysis.pagerank(kn)\n",
    "pr_vector = list(pr.values())\n",
    "pr_vector = jnp.array(pr_vector) * 3000\n",
    "nx.draw_networkx(kn, node_size=pr_vector, node_color=\"red\", labels=None)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Computations of eigenvalues\n",
    "\n",
    "- How to compute  eigenvalues and eigenvectors? \n",
    "\n",
    "There are two types of eigenproblems:\n",
    "\n",
    "- full eigenproblem (all eigenvalues & eigenvectors are required)\n",
    "- partial eigenvalues (minimal/maximal eigenvalues, eigenvalues within the specified region are required)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Computation of the eigenvalues via characteristic equations\n",
    "The eigenvalue problem has the form  \n",
    "\n",
    "$$ Ax = \\lambda x, $$\n",
    "\n",
    "or\n",
    "\n",
    "$$ (A - \\lambda I) x = 0, $$\n",
    "\n",
    "therefore matrix $A - \\lambda I$ has non-trivial kernel and should be singular. \n",
    "\n",
    "That means, that the **determinant**  \n",
    "\n",
    "$$ p(\\lambda) = \\det(A - \\lambda I) = 0. $$\n",
    "\n",
    "- The equation is called **characteristic equations** and is a polynomial of order $n$. \n",
    "- The $n$-degree polynomial has $n$ complex roots!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Recall the determinant\n",
    "The determinant of a square matrix $A$ is defined as \n",
    "\n",
    "$$\\det A = \\sum_{\\sigma \\in S_n} \\mathrm{sgn}({\\sigma})\\prod^n_{i=1} a_{i, \\sigma_i},$$\n",
    "\n",
    "where \n",
    "- $S_n$ is the set of all **permutations** of the numbers $1, \\ldots, n$\n",
    "- $\\mathrm{sgn}$ is the **signature** of the permutation ( $(-1)^p$, where $p$ is the number of transpositions to be made)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Properties of determinant\n",
    "Determinant has many nice properties:\n",
    "\n",
    "_1._ $\\det(AB) = \\det(A) \\det(B)$\n",
    "\n",
    "_2._ If we have one row as a sum of two vectors, determinant is a sum of two determinants\n",
    "\n",
    "_3._ \"Minor expansion\": we can expand  determinant through a selected row or column.\n",
    "\n",
    "- If you do it via **minor expansion**, we get **exponential** complexity in $n$.\n",
    "\n",
    "- Can we do $\\mathcal{O}(n^3)$? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Eigenvalues and characteristic equation\n",
    "\n",
    "- Now we go back to the eigenvalues.\n",
    "\n",
    "- The characteristic equation can be used to compute the eigenvalues, which leads to **na&iuml;ve** algorithm:\n",
    "\n",
    "$$p(\\lambda) = \\det(A - \\lambda I)$$\n",
    "\n",
    "1. Compute coefficients of the polynomial\n",
    "2. Compute the roots\n",
    "\n",
    "**Is this a good idea**?   \n",
    "\n",
    "**Give your feedback**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "We can do a short demo of this"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Text(0, 0.5, 'Imaginary part')"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as jnp\n",
    "import jax\n",
    "n = 40\n",
    "a = [[1.0 / (i - j + 0.5) for i in range(n)] for j in range(n)]\n",
    "a = jnp.array(a)\n",
    "ev = jnp.linalg.eigvals(a)\n",
    "#There is a special numpy function for chacteristic polynomial\n",
    "cf = jnp.poly(a)\n",
    "ev_roots = jnp.roots(cf)\n",
    "#print('Coefficients of the polynomial:', cf)\n",
    "#print('Polynomial roots:', ev_roots)\n",
    "plt.scatter(ev_roots.real, ev_roots.imag, marker='x', label='roots')\n",
    "b = a + 0 * jax.random.normal(jax.random.PRNGKey(0,), (n, n))\n",
    "ev_b = jnp.linalg.eigvals(b)\n",
    "plt.scatter(ev_b.real, ev_b.imag, marker='o', label='Lapack')\n",
    "#plt.scatter(ev_roots.real, ev_roots.imag, marker='o', label='Brute force')\n",
    "plt.legend(loc='best')\n",
    "plt.xlabel('Real part')\n",
    "plt.ylabel('Imaginary part')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Morale\n",
    "\n",
    "- Do not do that, unless you have a reason. \n",
    "- Polynomial rootfinding is very **ill-conditioned** (can be much better, but not with monomials $\\{1,x,x^2,\\dots\\}$!). Note that Gram matrix of monomials \n",
    "\n",
    "$$h_{ij} = \\int_0^1 x^i x^j\\, dx = \\frac{1}{i+j+1},$$\n",
    "\n",
    "is the Hilbert matrix, which has exponential decay of singular values. \n",
    "- So, monomials are \"almost\" linearly dependent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Gershgorin circles\n",
    "- There is a very interesting theorem that sometimes helps to localize the eigenvalues. \n",
    "\n",
    "- It is called **Gershgorin theorem**.  \n",
    "\n",
    "- It states that all eigenvalues $\\lambda_i,  i = 1, \\ldots, n$ are located inside the union of **Gershgorin circles** $C_i$,  where $C_i$ is a disk on the complex plane with center $a_{ii}$ and radius  \n",
    "\n",
    "$$r_i = \\sum_{j \\ne i} |a_{ij}|.$$\n",
    "\n",
    "- Moreover, if the circles do not intersect they contain only one eigenvalue per circle. \n",
    "- The proof is instructive since it uses the concepts we looked at the previous lectures."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Proof\n",
    "First, we need to show that if the matrix $A$ is **strictly diagonally dominant**, i.e. \n",
    "\n",
    "$$\n",
    "   |a_{ii}| > \\sum_{j \\ne i} |a_{ij}|,\n",
    "$$\n",
    "then such matrix is non-singular."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We separate the diagonal part and off-diagonal part, and get  \n",
    "\n",
    "$$\n",
    "    A = D + S = D( I + D^{-1}S),\n",
    "$$\n",
    "\n",
    "and $\\Vert D^{-1} S\\Vert_1 < 1$. Therefore, by using the **Neumann series**, the matrix $I + D^{-1}S$ is invertible and hence $A$ is invertible."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Now the proof follows by contradiction: \n",
    "\n",
    "- if any of the eigenvalues lies outside all of the circles, the matrix $(A - \\lambda I)$ \n",
    "is strictly diagonally dominant\n",
    "- thus it is invertible\n",
    "- that means, that $(A - \\lambda I) x = 0$ means $x = 0$. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## A short demo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import jax.numpy as jnp\n",
    "%matplotlib inline\n",
    "n = 3\n",
    "fig, ax = plt.subplots(1, 1)\n",
    "a = [[5, 1, 1], [1, 0, 0.5], [2, 0, 10]]\n",
    "#a = [[1.0 / (i - j + 0.5) for i in xrange(n)] for j in xrange(n)]\n",
    "a = jnp.array(a)\n",
    "#a = np.diag(np.arange(n))\n",
    "a = a + 2 * jax.random.normal(jax.random.PRNGKey(0), (n, n))\n",
    "#u = np.random.randn(n, n)\n",
    "#a = np.linalg.inv(u).dot(a).dot(u)\n",
    "xg = jnp.diag(a).real\n",
    "yg = jnp.diag(a).imag\n",
    "rg = jnp.zeros(n)\n",
    "ev = jnp.linalg.eigvals(a)\n",
    "for i in range(n):\n",
    "    rg = jax.ops.index_update(rg, jax.ops.index[i], jnp.sum(jnp.abs(a[i, :])) - jnp.abs(a[i, i]))\n",
    "    crc = plt.Circle((xg[i], yg[i]), radius=rg[i], fill=False)\n",
    "    ax.add_patch(crc)\n",
    "plt.scatter(ev.real, ev.imag, color='r', label=\"Eigenvalues\")\n",
    "plt.axis('equal')\n",
    "plt.legend()\n",
    "ax.set_title('Eigenvalues and Gershgorin circles')\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "**Note**: There are more complicated figures, like **[Cassini ovals](https://en.wikipedia.org/wiki/Cassini_oval)**, that include the spectrum\n",
    "\n",
    "$$ _{ij} = \\{z\\in\\mathbb{C}: |a_{ii} - z|\\cdot |a_{jj} - z|\\leq r_i r_j\\}, \\quad r_i = \\sum_{l\\not= i} |a_{il}|. $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Power method\n",
    "\n",
    "- We are often interested in the computation of the part of the spectrum, like the largest eigenvalues or smallest eigenvalues. \n",
    "- Also it is interesting to note that for the Hermitian matrices $(A = A^*)$ the eigenvalues are always real (prove it!).  \n",
    "- A power method is the simplest method for the computation of **the largest eigenvalue in modulus**. \n",
    "- It is also our first example of the **iterative method** and **Krylov method**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Power method\n",
    "\n",
    "- The eigenvalue problem\n",
    "\n",
    "$$Ax = \\lambda x, \\quad \\Vert x \\Vert_2 = 1 \\ \\text{for stability}.$$ \n",
    "\n",
    "can be rewritten as a **fixed-point iteration**.\n",
    "- This iteration is called **power method** and finds the largest in modulus eigenvalue of $A$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Power method has the form\n",
    "\n",
    "$$ x_{k+1} = A x_k, \\quad x_{k+1} := \\frac{x_{k+1}}{\\Vert x_{k+1} \\Vert_2}$$\n",
    "\n",
    "and \n",
    "\n",
    "$$ x_{k+1}\\to v_1,$$ \n",
    "\n",
    "where $Av_1 = \\lambda_1 v_1$ and $\\lambda_1$ is the largest eigenvalue and $v_1$ is the corresponding eigenvector.\n",
    "\n",
    "- On the $(k+1)$-th iteration approximation to $\\lambda_1$ can be found as\n",
    "\n",
    "$$ \\lambda^{(k+1)} = (Ax_{k+1}, x_{k+1}), $$\n",
    "\n",
    "- Note that $\\lambda^{(k+1)}$ is not required for the $(k+2)$-th iteration, but might be useful to measure error on each iteration: $\\|Ax_{k+1} - \\lambda^{(k+1)}x_{k+1}\\|$. \n",
    "\n",
    "- The convergence is geometric, but the convergence ratio is $q^k$, where $q = \\left|\\frac{\\lambda_{2}}{\\lambda_{1}}\\right| < 1$, for $\\lambda_1>\\lambda_2\\geq\\dots\\geq \\lambda_n$ and $k$ is the number of iteration. \n",
    "- It means, the convergence can be artitrary small. To prove it, it is sufficient to consider a $2 \\times 2$ diagonal matrix."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Convergence analysis for $A=A^*$\n",
    "\n",
    "Let's have a more precise look at the power method when $A$ is Hermitian.\n",
    "In two slides you will learn that every Hermitian matrix is diagonalizable. Therefore, there exists orthonormal basis of eigenvectors $v_1,\\dots,v_n$ such that $Av_i = \\lambda_i v_i$. Let us decompose $x_0$ into a sum of $v_i$ with coefficients $c_i$:\n",
    "\n",
    "$$ x_0 = c_1 v_1 + \\dots + c_n v_n. $$\n",
    "\n",
    "Since $v_i$ are eigenvectors, we have\n",
    "\n",
    "$$\n",
    "\\begin{split}\n",
    "x_1 &= \\frac{Ax_0}{\\|Ax_0\\|} = \\frac{c_1 \\lambda_1 v_1 + \\dots + c_n \\lambda_n v_n}{\\|c_1 \\lambda_1 v_1 + \\dots + c_n \\lambda_n v_n \\|}  \\\\\n",
    "&\\vdots\\\\\n",
    "x_k &= \\frac{Ax_{k-1}}{\\|Ax_{k-1}\\|} = \\frac{c_1 \\lambda_1^k v_1 + \\dots + c_n \\lambda_n^k v_n}{\\|c_1 \\lambda_1^k v_1 + \\dots + c_n \\lambda_n^k v_n \\|}\n",
    "\\end{split}\n",
    "$$\n",
    "\n",
    "Now you see, that \n",
    "\n",
    "$$\n",
    "x_k = \\frac{c_1}{|c_1|}\\left(\\frac{\\lambda_1}{|\\lambda_1|}\\right)^k\\frac{ v_1 + \\frac{c_2}{c_1}\\frac{\\lambda_2^k}{\\lambda_1^k}v_2 + \\dots + \\frac{c_n}{c_1}\\frac{\\lambda_n^k}{\\lambda_1^k}v_n}{\\left\\|v_1 + \\frac{c_2}{c_1}\\frac{\\lambda_2^k}{\\lambda_1^k}v_2 + \\dots + \\frac{c_n}{c_1}\\frac{\\lambda_n^k}{\\lambda_1^k}v_n\\right\\|},\n",
    "$$\n",
    "\n",
    "which converges to $v_1$ since $\\left| \\frac{c_1}{|c_1|}\\left(\\frac{\\lambda_1}{|\\lambda_1|}\\right)^k\\right| = 1$ and $\\left(\\frac{\\lambda_2}{\\lambda_1}\\right)^k \\to 0$ if $|\\lambda_2|<|\\lambda_1|$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Things to remember about the power method\n",
    "- Power method gives estimate of the largest eigenvalue in modulus or spectral radius of the given matrix\n",
    "- One step requires one matrix-by-vector product. If the matrix allows for an $\\mathcal{O}(n)$ matvec (for example, it is sparse), \n",
    "  then power method is tractable for larger $n$.\n",
    "- Convergence can be slow\n",
    "- If only a rough estimate is needed, only a few iterations are sufficient\n",
    "- The solution vector is in the **Krylov subspace** $\\{x_0, Ax_0,\\dots,A^{k}x_0\\}$ and has the form $\\mu A^k x_0$, where $\\mu$ is the normalization constant. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "#  Matrix decomposition: the Schur form\n",
    "There is one class of matrices when eigenvalues can be found easily: **triangular matrices**\n",
    "\n",
    "$$\n",
    "  A = \\begin{pmatrix}\n",
    "   \\lambda_1 & * & * \\\\\n",
    "   0 & \\lambda_2 & * \\\\\n",
    "   0 & 0 & \\lambda_3 \\\\\n",
    "  \\end{pmatrix}.\n",
    "$$\n",
    "\n",
    "The eigenvalues of $A$ are $\\lambda_1, \\lambda_2, \\lambda_3$. Why? "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Because the determinant is  \n",
    "\n",
    "$$\n",
    "   \\det(A - \\lambda I) = (\\lambda - \\lambda_1) (\\lambda - \\lambda_2) (\\lambda - \\lambda_3).\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "- Thus, computing the eigenvalues of triangular matrices is easy. Now, the unitary matrices come to help. \n",
    "- Let $U$ be a unitary matrix, i.e. $U^* U = I$.  Then \n",
    "\n",
    "$$\n",
    "   \\det(A - \\lambda I) = \\det(U (U^* A U - \\lambda I) U^*) = \\det(UU^*) \\det(U^* A U - \\lambda I) = \\det(U^* A U - \\lambda I),\n",
    "$$\n",
    "\n",
    "where we have used the famous multiplicativity property of the determinant, $\\det(AB) = \\det(A) \\det(B)$. \n",
    "- It means, that the matrices $U^* A U$ and $A$ have the same characteristic polynomials, and the same eigenvalues. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- If we manage to make $U^* A U = T$ where $T$ is **upper triangular**, then we are done.  \n",
    "\n",
    "- Multplying from the left and the right by $U$ and $U^*$ respectively, we get the desired decomposition:  \n",
    "\n",
    "$$ A = U T U^*. $$\n",
    "\n",
    "This is the celebrated **Schur decomposition**. \n",
    "- Recall that unitary matrices imply stability, thus the eigenvalues are computed very accurately. The Schur decomposition shows why we need matrix decompositions: it represents a matrix into a product of three matrices with a convenient structure.  "
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    "## Schur theorem\n",
    "\n",
    "**Theorem:** Every $A \\in \\mathbb{C}^{n \\times n}$ matrix can be represented in the Schur form $A = UTU^*$, where $U$ is unitary and $T$ is upper triangular.  \n",
    "\n",
    "**Sketch of the proof**.\n",
    "1. Every matrix has at least $1$ non-zero eigenvector (take a root of characteristic polynomial, $(A-\\lambda I)$ is singular, has    non-trivial nullspace). Let \n",
    "\n",
    "$$Av_1 = \\lambda_1 v_1, \\quad \\Vert v_1 \\Vert_2 = 1$$\n",
    "\n",
    "2. Let $U_1 = [v_1,v_2,\\dots,v_n]$, where $v_2,\\dots, v_n$ are any vectors othogonal to $v_1$. Then \n",
    "  \n",
    "$$ U^*_1 A U_1 = \\begin{pmatrix} \\lambda_1 & *  \\\\ 0 & A_2  \\end{pmatrix}, $$\n",
    "  \n",
    "where $A_2$ is an $(n-1) \\times (n-1)$ matrix. This is called **block triangular form**. We can now work with $A_2$ only and so on.  \n",
    "  \n",
    "**Note**: Since we need eigenvectors in this proof, this proof is not a practical algorithm."
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    "## Application of the Schur theorem\n",
    "\n",
    "- Important application of the Schur theorem:  **normal matrices**.  \n",
    "\n",
    "- **Definition.** Matrix $A$ is called **normal matrix**, if  \n",
    "\n",
    "$$ AA^* = A^* A. $$\n",
    "\n",
    "**Q:** Examples of normal matrices?"
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    "Examples: Hermitian matrices, unitary matrices.  "
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    "## Normal matrices\n",
    "\n",
    "**Theorem**: $A$ is a **normal matrix**, iff $A = U \\Lambda U^*$, where $U$ is unitary and $\\Lambda$ is diagonal. \n",
    "\n",
    "**Sketch of the proof:** \n",
    "- One way is straightforward (if the decomposition holds, the matrix is normal).  \n",
    "- The other is more complicated. Consider the Schur form of the matrix $A$. Then $AA^* = A^*A$ means $TT^* = T^* T$.  \n",
    "- By looking at the elements we immediately see, that the only upper triangular matrix $T$ that satisfies $TT^* = T^* T$ is a diagonal matrix!\n",
    "\n",
    "#### Important consequence\n",
    "\n",
    "Therefore, every normal matrix is **unitary diagonalizable**, which means that it can be diagonalized by unitary matrix $U$. \n",
    "\n",
    "In other words every normal matrix has orthogonal basis of eigenvectors."
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    "## How we compute the Schur decomposition?\n",
    "\n",
    "- Everything is fine, but how we compute the Schur form?\n",
    "\n",
    "- This will be covered in the next lecture."
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    "## Variational principle for eigenvalues\n",
    "\n",
    "- In many cases, minimal/maximal eigenvalues are needed\n",
    "- Then, if $A$ is a Hermitian matrix, the **Rayleigh quotient** is defined as\n",
    "\n",
    "$$R_A(x) = \\frac{(Ax, x)}{(x, x)},$$\n",
    "\n",
    "and the maximal eigenvalue is the maximum of $R_A(x)$, and the minimal eigenvalue is the minimal of $R_A(x)$. \n",
    "\n",
    "- Thus, we can use optimization method to find these **extreme eigenvalues**.\n",
    "\n",
    "Now, \"advanced\" concept."
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    "## Spectrum and pseudospectrum\n",
    "\n",
    "- For linear dynamical systems given by the matrix $A$, spectrum can tell a lot about the system (i.e. stability, ...)\n",
    "\n",
    "- However, for **non-normal matrices**, spectrum can be unstable with respect to small perturbations.\n",
    "\n",
    "- In order to measure such perturbation, the notion of **pseudospectrum** has been developed."
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    "## Pseudospectrum\n",
    "\n",
    "We consider the union of all possible eigenvalues of all perturbations of the matrix $A$.\n",
    "\n",
    "$$\\Lambda_{\\epsilon}(A) = \\{ \\lambda \\in \\mathbb{C}: \\exists E, x \\ne 0: (A + E) x = \\lambda x, \\quad \\Vert E \\Vert_2 \\leq \\epsilon. \\}$$\n",
    "\n",
    "- For small $E$ and normal $A$ these will be circules around eigenvalues, for non-normal matrices, the structure can be much different. More details: http://www.cs.ox.ac.uk/pseudospectra/\n",
    "\n",
    "<img src='pseudospectrum.gif' width=700>"
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   "source": [
    "## Summary of todays lecture\n",
    "\n",
    "- Eigenvalues, eigenvectors\n",
    "- Gershgorin theorem\n",
    "- Power method\n",
    "- Schur theorem \n",
    "- Normal matrices\n",
    "- Some advanced topics\n",
    "\n"
   ]
  },
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   "source": [
    "## Next lecture\n",
    "- Matrix factorizations and how we compute them."
   ]
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   "source": [
    "### Questions?"
   ]
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    "from IPython.core.display import HTML\n",
    "def css_styling():\n",
    "    styles = open(\"./styles/custom.css\", \"r\").read()\n",
    "    return HTML(styles)\n",
    "css_styling()"
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