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yunruse/concat.ipynb

Created March 20, 2023 21:24
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Concatenative 2x2 matrices
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concat.ipynb
{
"cells": [
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"# Counting concatenative matrices\n",
"\n",
"$$\n",
"\\left(\\begin{matrix}\n",
"\\color{red}3 & \\color{red}4\\\\\n",
"\\color{red}8 & \\color{red}7\n",
"\\end{matrix}\\right)\n",
"\\left(\\begin{matrix}\n",
"\\color{green}7 & \\color{green}2\\\\\n",
"\\color{green}4 & \\color{green}9\n",
"\\end{matrix}\\right)\n",
"= \n",
"\\left(\\begin{matrix}\n",
"\\color{red}3\\color{green}7 & \\color{red}4\\color{green}2\\\\\n",
"\\color{red}8\\color{green}4 & \\color{red}7\\color{green}9\n",
"\\end{matrix}\\right)\n",
"$$\n",
"\n",
"This is a fun little trick. We'll say these two matrices are *concatenative*.\n",
"\n",
"That is to say: two square matrices $A, B$ of the same size are *concatenative* in base $b$ if $AB=A\\star_b B$ for some elementwise concatenation operator $(A\\star_b B)_{ij} = bA_{ij}+B_{ij}$.\n",
"\n",
"Now, let us ask: for two matrices of size $N$ and base $b$, how *many* pairs of concatenative matrices exist? Evidently there are $b^{2N^2}$ such pairs of matrices - it will quickly become impractical to brute-force."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's start with some imports and define the above:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"from sympy import Symbol, Matrix, init_printing\n",
"init_printing(use_unicode=True)\n",
"import numpy as np\n",
"import sympy as sy"
]
},
{
"cell_type": "code",
"execution_count": 85,
"metadata": {},
"outputs": [],
"source": [
"def concatenate(base: int):\n",
" def ufunc(a, b):\n",
" return a*base + b\n",
" return np.vectorize(ufunc)\n",
"\n",
"def is_concatenative(base, a, b):\n",
" return a@b == Matrix(concatenate(base)(a, b))"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Symbolic matrices\n",
"Let's symbolically represent a matrix\n",
"$\n",
"A = \n",
"\\left(\\begin{matrix}\n",
"A_{11} & A_{12} \\\\\n",
"A_{21} & A_{22}\n",
"\\end{matrix}\\right)\n",
"$:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}A_{0,0} & A_{1,0}\\\\A_{0,1} & A_{1,1}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡A_{0,0} A_{1,0}⎤\n",
"⎢ ⎥\n",
"⎣A_{0,1} A_{1,1}⎦"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def symbolic_matrix(name: str, size: int):\n",
" return Matrix(\n",
" [[\n",
" Symbol('%s_{%s,%s}' % (name, i, j), positive=True, integer=True)\n",
" for i in range(size)] for j in range(size)]\n",
" )\n",
"\n",
"symbolic_matrix('A', 2)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"Neat! Now we need only solve the equation $AB-A\\star_b B = 0$:"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"def solvable(size: int):\n",
" A = symbolic_matrix('A', size)\n",
" B = symbolic_matrix('B', size)\n",
" base = Symbol('b', positive=True, integer=True)\n",
" return A, B, base, (A@B) - concatenate(base)(A, B)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"It's certainly not the prettiest, but we know that each element must equal zero:"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}A_{0,0} B_{0,0} - A_{0,0} n + A_{1,0} B_{0,1} - B_{0,0} & A_{0,0} B_{1,0} + A_{1,0} B_{1,1} - A_{1,0} n - B_{1,0}\\\\A_{0,1} B_{0,0} - A_{0,1} n + A_{1,1} B_{0,1} - B_{0,1} & A_{0,1} B_{1,0} + A_{1,1} B_{1,1} - A_{1,1} n - B_{1,1}\\end{matrix}\\right]$"
],
"text/plain": [
"⎡A_{0,0}⋅B_{0,0} - A_{0,0}⋅n + A_{1,0}⋅B_{0,1} - B_{0,0} A_{0,0}⋅B_{1,0} + A_\n",
"⎢ \n",
"⎣A_{0,1}⋅B_{0,0} - A_{0,1}⋅n + A_{1,1}⋅B_{0,1} - B_{0,1} A_{0,1}⋅B_{1,0} + A_\n",
"\n",
"{1,0}⋅B_{1,1} - A_{1,0}⋅n - B_{1,0}⎤\n",
" ⎥\n",
"{1,1}⋅B_{1,1} - A_{1,1}⋅n - B_{1,1}⎦"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"solvable(2)[2]"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"We can solve this equation for $A$:"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"from sympy.solvers import solve"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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uXdDt3FhZsXY7bVVFR2MujdU5dLTdK9PsIcxcvnBkyk6qjxUz+yrw1SH/3hG4B/jViMc3nQUkTwPcDuDubwAOB/YA3tcBe2WJ7W8vfT8eOJZkvdCzgcvcfa6ZBW0ulkHMcrWtzaG77Z5rx5PdqZcBPzazCyuyWykFNHeobpvZdcCs8XgWDfXfbGL63Evfu6TZddirgi62e66dlmj2dWTrrmLtdtsrS1djLo3V2XS13XPtNF23C8bKirXbaasquhhzaazOp4vtnmunpGZXNhfe6En1HA4CPlFhozWNBcA6wG/cHWADknWRDjGzcWwKENteWWL72yN5zO/Q6bv+7n4Sye7sc4H7KrITs1xta3PobrsXsfNu4AjgDnc/OD3uQ2Z2Z0U+xEC63U5bVRHT5x7d0+w67FVBF9u9iB1pdvNR/82mqzGXxupsutruRexIt5uP+m82XYy5NFbn08V2L2KnEZpdZlL9lcq8GAEz26NO+xGYD1wBfDb9vAXJ41kXuPs8M3sBwN3fS3LHZjbwRTP7+rAMc9Lm2nP3K0kW9L/GzA7LK0CR9O5+OnBqTlb7pnfW+4ldP7sDV8x4jG56Y5p1KirTuMo1rB2K2irc7kX9itDuIT5npQ1p91NI1jGbZj1gyt1P6PvuQDO7YYAbuXbM7EaSem0t0m17IaTvQmafit0XptOMVd8qrJ/CfbdkuWJqdiF7I4zVub5FaPfomp3mNapuS7O7gWLt11GsrVh7nD4r1o6EdFuxdp7PirXLx9oF8phpT7H26zQt1p4qeXwpB/6vrHGRyXzgJjO7P339DDgT2AHYGcDd1yXZIGU/kpPuRE/WtlyLAmlz7QHnAkcGlKFI+qXAvJzXoLWZYtdPD/jFjMMWAs8B91dUpsrLlTKsHYq0edbxaxDo11LG1O4hPhdI26N4u5+fpp9+XTXgu1srsCOaS+b5Gdp3c9LH7gvTLGVM+lZx/fQI61OjliumZheyl3P8GgT4tpQxjmshPuek7RHW7qPqdqgd0UwUayvWVqw9Zp8LpO2hWFsUR7G2Yu0mxNp5ebyGYu3Gx9ovls2gzC/VV+YnEaPg7jsCmwO3zfjXW9P3X6fv7wLuNrPl6XFXAwcA3xqQ7dC0Re2Z2XXuvqhoOYqkN7MngCeK5gm11M+GwNvpu/PmyfpNxwMXmdlLZcs0xnINbIcAWyHtXtivMbd70Lk6LO0I7b4KWNWX9llglZllin6oHdFMCp6fQX13WHp3/2kBW0A1fWFGmnHqW1X18z0C+1SDxqIsTRrHWF3It3GPa3VodppXsG5Ls7uBYu1sFGsr1q7QZ8XaohIUa2ejWDterJ2VxwAUazc71n6mbAZlJtXvAdZ199lm9mpZR8QaLEjfV7r71sCGJLvdnglcbGbTG49sCyzvO245MGdInllpi9prCrHrZ1fgVeBId7+WROyWANsBB5cpyAzGUa6ytkKowq8sYp+nsdo9lh0xXnLPT3dfSFgfGdan2qbZELd+uqjZIfZC6JJuS7NFCIq1s1GsrVhbsbZoGoq1s1Gsvbaf47IXQpd0u2ua/SrwYNlMykyq/zvJYwzbV+GIWIPpjnFv+v40ySMOnwf+oQP2yhLb3x7wAMkaV5cCWwJXA3umdxSrIma52tbm0N12j2VHjBf132xi+tyje5pdh70q6GK7x7Ijxov6bzZdjbk0VmfT1XaPZUeMF/XfbLoYc2mszqeL7R7DzrYkTxA8XzajkSfVzewed/8x8GHgr8o6Il7HzE4GTi6Q9DHWvMs1h+HrLA1NG2CvEdRQP+eTrPsEyQYQY2FM5SprK4TSfmUR+zwt2+5mtjiGHdEMCp6foX1kYHozO6uArUYRuX6+Scc0O9BeCJ3R7Sq0tIhuS7O7gWLtbBRrK9aOgWJtEYJi7WwUayvWHjddibXdfTawD3BRFfmV+aU6wMeAG939DjP7ThUOiSBuAd7h7nNI7kgdCJwG4O7XAEdOr92UlbYKBthrAo2pn4oJKdfYaGMdNvQ8FZNDZh+J2aca2hcaUz8V0wjNHmKv0XXY0PNUTBaNiSUb2h8aUz8V0wjdbmMdNvQ8FZNDY2LJhvaFxtRPxTRCs4fYa3QdNvQ8HTdHAb8LnFdFZrOmpqbW+CJdkP5aYG8zuzEvA3ffCbgAeNnM9q/CKVEcd38/cBYwG/iSmV2QLuL/ELBT/+MMg9IG2voRsBuwEclmA4eb2c0Z9gamH7WsoxCzfmISWK5S7TDoeOCnQ2w1og4Dfa79PBWTw7A+ElO3aXBfiF0/sahbs3PG6trrsG3nqZgsFGvn+qxYW7G2dFs0BsXauf4q1q6gHdqm2207T8eBu68LHAIsBT5gZjcUPG4RGXPkgybVFwLfBlYAL6Zfn2RmmY8ouPv2ZvZwEafEeHH3ecBHzOy4LtorS9v8LUrMcrWxDtvos5gc1H+zaaPPeWiszqZt/orJQv03m7b5WxSN1dm00WcxOaj/ZtNGn/PQWJ1N2/wti7t/iGS99q/kzV27+17AGenH9YGtgcPM7NaZadeaVBdCCCGEEEIIIYQQQgghxGBm1+2AEEIIIYQQQgghhBBCCNEW/h+fgo+VOMHf8QAAAABJRU5ErkJggg==",
"text/latex": [
"$\\displaystyle \\left[ \\left\\{ A_{0,0} : - \\frac{- B_{0,0} B_{1,1} + B_{0,0} b + B_{0,1} B_{1,0}}{B_{0,0} B_{1,1} - B_{0,0} b - B_{0,1} B_{1,0} - B_{1,1} b + b^{2}}, \\ A_{0,1} : - \\frac{B_{0,1} b}{B_{0,0} B_{1,1} - B_{0,0} b - B_{0,1} B_{1,0} - B_{1,1} b + b^{2}}, \\ A_{1,0} : - \\frac{B_{1,0} b}{B_{0,0} B_{1,1} - B_{0,0} b - B_{0,1} B_{1,0} - B_{1,1} b + b^{2}}, \\ A_{1,1} : - \\frac{- B_{0,0} B_{1,1} + B_{0,1} B_{1,0} + B_{1,1} b}{B_{0,0} B_{1,1} - B_{0,0} b - B_{0,1} B_{1,0} - B_{1,1} b + b^{2}}\\right\\}\\right]$"
],
"text/plain": [
"⎡⎧ -(-B_{0,0}⋅B_{1,1} + B_{0,0}⋅b + B_{0,1}⋅B_{1,0}) \n",
"⎢⎪A_{0,0}: ──────────────────────────────────────────────────────────────, A_{\n",
"⎢⎨ 2 \n",
"⎢⎪ B_{0,0}⋅B_{1,1} - B_{0,0}⋅b - B_{0,1}⋅B_{1,0} - B_{1,1}⋅b + b \n",
"⎣⎩ \n",
"\n",
" -B_{0,1}⋅b \n",
"0,1}: ──────────────────────────────────────────────────────────────, A_{1,0}:\n",
" 2 \n",
" B_{0,0}⋅B_{1,1} - B_{0,0}⋅b - B_{0,1}⋅B_{1,0} - B_{1,1}⋅b + b \n",
" \n",
"\n",
" -B_{1,0}⋅b \n",
" ──────────────────────────────────────────────────────────────, A_{1,1}: ────\n",
" 2 \n",
" B_{0,0}⋅B_{1,1} - B_{0,0}⋅b - B_{0,1}⋅B_{1,0} - B_{1,1}⋅b + b B_{0\n",
" \n",
"\n",
" -(-B_{0,0}⋅B_{1,1} + B_{0,1}⋅B_{1,0} + B_{1,1}⋅b) ⎫⎤\n",
"──────────────────────────────────────────────────────────⎪⎥\n",
" 2⎬⎥\n",
",0}⋅B_{1,1} - B_{0,0}⋅b - B_{0,1}⋅B_{1,0} - B_{1,1}⋅b + b ⎪⎥\n",
" ⎭⎦"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A, B, base_symbol, solv = solvable(2)\n",
"solns = solve(solv)[0]\n",
"solns"
]
},
{
"cell_type": "code",
"execution_count": 122,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}3.0 & 4.0\\\\8.0 & 7.0\\end{matrix}\\right]$"
],
"text/plain": [
"⎡3.0 4.0⎤\n",
"⎢ ⎥\n",
"⎣8.0 7.0⎦"
]
},
"execution_count": 122,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"def get_a(base: int, my_b: Matrix) -> Matrix:\n",
" b_map = {base_symbol: base} | dict(zip(B, my_b))\n",
" return A.evalf(subs={a: v.evalf(subs=b_map) for a, v in solns.items()})\n",
"\n",
"get_a(10, Matrix([[7, 2], [4, 9]]))"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"Just because `get_a` works for a clear example doesn't mean it will always return clean values, however!"
]
},
{
"cell_type": "code",
"execution_count": 123,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}-9.0 & -11.25\\\\0 & -0.25\\end{matrix}\\right]$"
],
"text/plain": [
"⎡-9.0 -11.25⎤\n",
"⎢ ⎥\n",
"⎣ 0 -0.25 ⎦"
]
},
"execution_count": 123,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"get_a(10, Matrix([[9, 9], [0, 2]]))"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"So let's iterate through every possible $B$ and find all working candidate $A$ matrices (where every $A_{ij}$ is an integer and $0 \\le A_ij \\lt b$):"
]
},
{
"cell_type": "code",
"execution_count": 139,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 0\n",
"1 1\n",
"2 1\n",
"3 2\n",
"4 5\n",
"5 8\n",
"6 21\n",
"7 22\n",
"8 54\n",
"9 54\n",
"10 101\n"
]
}
],
"source": [
"from itertools import product\n",
"def iterate_matrix(base: int):\n",
" for w, x, y, z in product(*[range(base)]*4):\n",
" b = Matrix([[w, x], [y, z]])\n",
" a = get_a(base, b)\n",
" # always a positive integer\n",
" if not all(i % 1 == 0 and 0 <= i < base for i in a):\n",
" continue\n",
" a = a.applyfunc(int)\n",
" # assert a@b == Matrix(concatenate(base)(a, b))\n",
" yield a, b\n",
"\n",
"BASES = {}\n",
"for base in range(11):\n",
" print(base, end=' ')\n",
" BASES[base] = list(iterate_matrix(base))\n",
" print(len(BASES[base]))"
]
},
{
"cell_type": "code",
"execution_count": 138,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right], \\ \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right], \\ \\left[\\begin{matrix}0 & 0\\\\0 & 0\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡0 0⎤ ⎡0 0⎤ ⎡0 0⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣0 0⎦ ⎣0 0⎦ ⎣0 0⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 2\\\\9 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 2\\\\9 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}22 & 22\\\\99 & 99\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 2⎤ ⎡2 2⎤ ⎡22 22⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣9 9⎦ ⎣9 9⎦ ⎣99 99⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 3\\\\6 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 3\\\\6 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}22 & 33\\\\66 & 99\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 3⎤ ⎡2 3⎤ ⎡22 33⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣6 9⎦ ⎣6 9⎦ ⎣66 99⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 6\\\\3 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 6\\\\3 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}22 & 66\\\\33 & 99\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 6⎤ ⎡2 6⎤ ⎡22 66⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣3 9⎦ ⎣3 9⎦ ⎣33 99⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 3\\\\3 & 5\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 6\\\\6 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}22 & 36\\\\36 & 58\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 3⎤ ⎡2 6⎤ ⎡22 36⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣3 5⎦ ⎣6 8⎦ ⎣36 58⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}5 & 6\\\\7 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 6\\\\7 & 6\\end{matrix}\\right], \\ \\left[\\begin{matrix}52 & 66\\\\77 & 96\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡5 6⎤ ⎡2 6⎤ ⎡52 66⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣7 9⎦ ⎣7 6⎦ ⎣77 96⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}5 & 7\\\\6 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 7\\\\6 & 6\\end{matrix}\\right], \\ \\left[\\begin{matrix}52 & 77\\\\66 & 96\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡5 7⎤ ⎡2 7⎤ ⎡52 77⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣6 9⎦ ⎣6 6⎦ ⎣66 96⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 9\\\\2 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}2 & 9\\\\2 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}22 & 99\\\\22 & 99\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 9⎤ ⎡2 9⎤ ⎡22 99⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣2 9⎦ ⎣2 9⎦ ⎣22 99⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 3\\\\8 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}3 & 3\\\\8 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}33 & 33\\\\88 & 88\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 3⎤ ⎡3 3⎤ ⎡33 33⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣8 8⎦ ⎣8 8⎦ ⎣88 88⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 4\\\\6 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}3 & 4\\\\6 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}33 & 44\\\\66 & 88\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 4⎤ ⎡3 4⎤ ⎡33 44⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣6 8⎦ ⎣6 8⎦ ⎣66 88⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}6 & 5\\\\9 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}3 & 5\\\\9 & 5\\end{matrix}\\right], \\ \\left[\\begin{matrix}63 & 55\\\\99 & 85\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡6 5⎤ ⎡3 5⎤ ⎡63 55⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣9 8⎦ ⎣9 5⎦ ⎣99 85⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 6\\\\4 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}3 & 6\\\\4 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}33 & 66\\\\44 & 88\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 6⎤ ⎡3 6⎤ ⎡33 66⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣4 8⎦ ⎣4 8⎦ ⎣44 88⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 8\\\\3 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}3 & 8\\\\3 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}33 & 88\\\\33 & 88\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 8⎤ ⎡3 8⎤ ⎡33 88⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣3 8⎦ ⎣3 8⎦ ⎣33 88⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}6 & 9\\\\5 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}3 & 9\\\\5 & 5\\end{matrix}\\right], \\ \\left[\\begin{matrix}63 & 99\\\\55 & 85\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡6 9⎤ ⎡3 9⎤ ⎡63 99⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣5 8⎦ ⎣5 5⎦ ⎣55 85⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 2\\\\8 & 7\\end{matrix}\\right], \\ \\left[\\begin{matrix}4 & 2\\\\8 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}24 & 22\\\\88 & 79\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 2⎤ ⎡4 2⎤ ⎡24 22⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣8 7⎦ ⎣8 9⎦ ⎣88 79⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}2 & 4\\\\4 & 7\\end{matrix}\\right], \\ \\left[\\begin{matrix}4 & 4\\\\4 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}24 & 44\\\\44 & 79\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡2 4⎤ ⎡4 4⎤ ⎡24 44⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣4 7⎦ ⎣4 9⎦ ⎣44 79⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}4 & 4\\\\7 & 7\\end{matrix}\\right], \\ \\left[\\begin{matrix}4 & 4\\\\7 & 7\\end{matrix}\\right], \\ \\left[\\begin{matrix}44 & 44\\\\77 & 77\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡4 4⎤ ⎡4 4⎤ ⎡44 44⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣7 7⎦ ⎣7 7⎦ ⎣77 77⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
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],
"text/plain": [
"⎛⎡7 2⎤ ⎡9 2⎤ ⎡79 22⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣8 2⎦ ⎣8 4⎦ ⎣88 24⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}9 & 2\\\\9 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 2\\\\9 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}99 & 22\\\\99 & 22\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡9 2⎤ ⎡9 2⎤ ⎡99 22⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣9 2⎦ ⎣9 2⎦ ⎣99 22⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 6\\\\4 & 3\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 3\\\\2 & 9\\end{matrix}\\right], \\ \\left[\\begin{matrix}39 & 63\\\\42 & 39\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 6⎤ ⎡9 3⎤ ⎡39 63⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣4 3⎦ ⎣2 9⎦ ⎣42 39⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}9 & 6\\\\6 & 3\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 3\\\\3 & 6\\end{matrix}\\right], \\ \\left[\\begin{matrix}99 & 63\\\\63 & 36\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡9 6⎤ ⎡9 3⎤ ⎡99 63⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣6 3⎦ ⎣3 6⎦ ⎣63 36⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 3\\\\4 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 3\\\\4 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}39 & 33\\\\44 & 28\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 3⎤ ⎡9 3⎤ ⎡39 33⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣4 2⎦ ⎣4 8⎦ ⎣44 28⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}6 & 3\\\\5 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 3\\\\5 & 5\\end{matrix}\\right], \\ \\left[\\begin{matrix}69 & 33\\\\55 & 25\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡6 3⎤ ⎡9 3⎤ ⎡69 33⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣5 2⎦ ⎣5 5⎦ ⎣55 25⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}9 & 3\\\\6 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 3\\\\6 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}99 & 33\\\\66 & 22\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡9 3⎤ ⎡9 3⎤ ⎡99 33⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣6 2⎦ ⎣6 2⎦ ⎣66 22⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}7 & 8\\\\4 & 3\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 4\\\\2 & 7\\end{matrix}\\right], \\ \\left[\\begin{matrix}79 & 84\\\\42 & 37\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡7 8⎤ ⎡9 4⎤ ⎡79 84⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣4 3⎦ ⎣2 7⎦ ⎣42 37⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 4\\\\3 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 4\\\\3 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}39 & 44\\\\33 & 28\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 4⎤ ⎡9 4⎤ ⎡39 44⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣3 2⎦ ⎣3 8⎦ ⎣33 28⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}7 & 4\\\\4 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 4\\\\4 & 4\\end{matrix}\\right], \\ \\left[\\begin{matrix}79 & 44\\\\44 & 24\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡7 4⎤ ⎡9 4⎤ ⎡79 44⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣4 2⎦ ⎣4 4⎦ ⎣44 24⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}6 & 5\\\\3 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 5\\\\3 & 5\\end{matrix}\\right], \\ \\left[\\begin{matrix}69 & 55\\\\33 & 25\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡6 5⎤ ⎡9 5⎤ ⎡69 55⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣3 2⎦ ⎣3 5⎦ ⎣33 25⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}3 & 6\\\\2 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 6\\\\2 & 8\\end{matrix}\\right], \\ \\left[\\begin{matrix}39 & 66\\\\22 & 28\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡3 6⎤ ⎡9 6⎤ ⎡39 66⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣2 2⎦ ⎣2 8⎦ ⎣22 28⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}9 & 6\\\\3 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 6\\\\3 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}99 & 66\\\\33 & 22\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡9 6⎤ ⎡9 6⎤ ⎡99 66⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣3 2⎦ ⎣3 2⎦ ⎣33 22⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}5 & 7\\\\2 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 7\\\\2 & 6\\end{matrix}\\right], \\ \\left[\\begin{matrix}59 & 77\\\\22 & 26\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡5 7⎤ ⎡9 7⎤ ⎡59 77⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣2 2⎦ ⎣2 6⎦ ⎣22 26⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}7 & 8\\\\2 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 8\\\\2 & 4\\end{matrix}\\right], \\ \\left[\\begin{matrix}79 & 88\\\\22 & 24\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡7 8⎤ ⎡9 8⎤ ⎡79 88⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣2 2⎦ ⎣2 4⎦ ⎣22 24⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left( \\left[\\begin{matrix}9 & 9\\\\2 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}9 & 9\\\\2 & 2\\end{matrix}\\right], \\ \\left[\\begin{matrix}99 & 99\\\\22 & 22\\end{matrix}\\right]\\right)$"
],
"text/plain": [
"⎛⎡9 9⎤ ⎡9 9⎤ ⎡99 99⎤⎞\n",
"⎜⎢ ⎥, ⎢ ⎥, ⎢ ⎥⎟\n",
"⎝⎣2 2⎦ ⎣2 2⎦ ⎣22 22⎦⎠"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"from IPython.display import display\n",
"for a, b in BASES[10]:\n",
" display((a, b, Matrix(concatenate(base)(a, b))))"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"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": "3.11.1"
},
"orig_nbformat": 4
},
"nbformat": 4,
"nbformat_minor": 2
}
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