{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Symbolic Tensor Notebook"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import sympy as sp\n",
"from IPython.display import Markdown\n",
"\n",
"\n",
"from mechpy.core.symbolic.tensor import (\n",
" SymbolicTensor,\n",
" SymbolicThreeByThreeTensor,\n",
" SymbolicSymmetricThreeByThreeTensor,\n",
" SymbolicSixBySixTensor,\n",
")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Symbolic Tensor"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"a, b, c, d, e, f, g, h, i = sp.symbols(\"a b c d e f g h i\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Init Method"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b], [c, d]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"data = sp.ImmutableDenseNDimArray([[a, b], [c, d]])\n",
"A = SymbolicTensor(data)\n",
"display(A.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### From List method"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Symbolic Tensor"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b], [c, d]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([a, b, c, d], shape=(2, 2))\n",
"display(A.data)"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c\\\\d & e & f\\\\g & h & i\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, c], [d, e, f], [g, h, i]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([a, b, c, d, e, f, g, h, i], shape=(3, 3))\n",
"display(A.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Symbolic 3x3 Tensor"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c\\\\d & e & f\\\\g & h & i\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, c], [d, e, f], [g, h, i]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicThreeByThreeTensor.from_list([a, b, c, d, e, f, g, h, i])\n",
"display(A.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Symbolic Symmetric 3x3 Tensor"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"list_notation_standard = [a, b, c, b, d, e, c, e, f]\n",
"list_notation_voigt = [a, f, e, f, b, d, e, d, c]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##### List 1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Default notation"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicSymmetricThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'standard'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list(list_notation_standard)\n",
"display(A.__class__)\n",
"display(A.data)\n",
"display(A.notation)"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c\\\\b & d & e\\\\c & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, c], [b, d, e], [c, e, f]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notation 2"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'voigt'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list(list_notation_voigt, notation=\"voigt\")\n",
"display(A.data)\n",
"display(A.notation)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & f & e\\\\f & b & d\\\\e & d & c\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, f, e], [f, b, d], [e, d, c]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"##### List 2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Default notation"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & f & e & b & d & c\\end{matrix}\\right]$"
],
"text/plain": [
"[a, f, e, b, d, c]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'standard'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list(list_notation_voigt)\n",
"display(A.data)\n",
"display(A.notation)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & f & e\\\\f & b & d\\\\e & d & c\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, f, e], [f, b, d], [e, d, c]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notation 2"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'voigt'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list(list_notation_voigt, notation=\"voigt\")\n",
"display(A.data)\n",
"display(A.notation)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & f & e\\\\f & b & d\\\\e & d & c\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, f, e], [f, b, d], [e, d, c]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### list of 6 elements"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"default notation"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'standard'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list([a, b, c, d, e, f])\n",
"display(A.data)\n",
"display(A.notation)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c\\\\b & d & e\\\\c & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, c], [b, d, e], [c, e, f]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Notation 2"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'voigt'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list([a, b, c, d, e, f], notation=\"voigt\")\n",
"display(A.data)\n",
"display(A.notation)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & f & e\\\\f & b & d\\\\e & d & c\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, f, e], [f, b, d], [e, d, c]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"#### Symbolic 6x6 Tensor"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"List 36 elements"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}1 & 2 & 3 & 4 & 5 & 6\\\\7 & 8 & 9 & 10 & 11 & 12\\\\13 & 14 & 15 & 16 & 17 & 18\\\\19 & 20 & 21 & 22 & 23 & 24\\\\25 & 26 & 27 & 28 & 29 & 30\\\\31 & 32 & 33 & 34 & 35 & 36\\end{matrix}\\right]$"
],
"text/plain": [
"[[1, 2, 3, 4, 5, 6], [7, 8, 9, 10, 11, 12], [13, 14, 15, 16, 17, 18], [19, 20, 21, 22, 23, 24], [25, 26, 27, 28, 29, 30], [31, 32, 33, 34, 35, 36]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"data_list = list(range(1,37))\n",
"A = SymbolicSixBySixTensor.from_list(data_list)\n",
"display(A.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"List 6x6 elemnts"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}1 & 2 & 3 & 4 & 5 & 6\\\\7 & 8 & 9 & 10 & 11 & 12\\\\13 & 14 & 15 & 16 & 17 & 18\\\\19 & 20 & 21 & 22 & 23 & 24\\\\25 & 26 & 27 & 28 & 29 & 30\\\\31 & 32 & 33 & 34 & 35 & 36\\end{matrix}\\right]$"
],
"text/plain": [
"[[1, 2, 3, 4, 5, 6], [7, 8, 9, 10, 11, 12], [13, 14, 15, 16, 17, 18], [19, 20, 21, 22, 23, 24], [25, 26, 27, 28, 29, 30], [31, 32, 33, 34, 35, 36]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"data_list = [[i * 6 + j + 1 for j in range(6)] for i in range(6)]\n",
"A = SymbolicSixBySixTensor.from_list(data_list)\n",
"display(A.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Create Method"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"This method is a simple way to create a tensor."
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}x_{11} & x_{12}\\\\x_{21} & x_{22}\\end{matrix}\\right]$"
],
"text/plain": [
"[[x_11, x_12], [x_21, x_22]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.create(shape=(2, 2), name=\"x\")\n",
"display(A.data)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}X_{11} & X_{12} & X_{13}\\\\X_{21} & X_{22} & X_{23}\\\\X_{31} & X_{32} & X_{33}\\end{matrix}\\right]$"
],
"text/plain": [
"[[X_11, X_12, X_13], [X_21, X_22, X_23], [X_31, X_32, X_33]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicThreeByThreeTensor.create(name=\"X\")\n",
"display(A.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Custom name"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} & a_{15} & a_{16}\\\\a_{21} & a_{22} & a_{23} & a_{24} & a_{25} & a_{26}\\\\a_{31} & a_{32} & a_{33} & a_{34} & a_{35} & a_{36}\\\\a_{41} & a_{42} & a_{43} & a_{44} & a_{45} & a_{46}\\\\a_{51} & a_{52} & a_{53} & a_{54} & a_{55} & a_{56}\\\\a_{61} & a_{62} & a_{63} & a_{64} & a_{65} & a_{66}\\end{matrix}\\right]$"
],
"text/plain": [
"[[a_11, a_12, a_13, a_14, a_15, a_16], [a_21, a_22, a_23, a_24, a_25, a_26], [a_31, a_32, a_33, a_34, a_35, a_36], [a_41, a_42, a_43, a_44, a_45, a_46], [a_51, a_52, a_53, a_54, a_55, a_56], [a_61, a_62, a_63, a_64, a_65, a_66]]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'a'"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSixBySixTensor.create(name=\"a\")\n",
"display(A.data)\n",
"display(A.name)"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6}\\end{matrix}\\right]$"
],
"text/plain": [
"[x_1, x_2, x_3, x_4, x_5, x_6]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'x'"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'standard'"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}x_{1} & x_{2} & x_{3}\\\\x_{2} & x_{4} & x_{5}\\\\x_{3} & x_{5} & x_{6}\\end{matrix}\\right]$"
],
"text/plain": [
"[[x_1, x_2, x_3], [x_2, x_4, x_5], [x_3, x_5, x_6]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.create(name=\"x\")\n",
"display(A.data)\n",
"display(A.name)\n",
"display(A.notation)\n",
"display(A.to_general().data)\n"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}x_{11} & x_{22} & x_{33} & x_{23} & x_{13} & x_{12}\\end{matrix}\\right]$"
],
"text/plain": [
"[x_11, x_22, x_33, x_23, x_13, x_12]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'x'"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"'voigt'"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}x_{11} & x_{12} & x_{13}\\\\x_{12} & x_{22} & x_{23}\\\\x_{13} & x_{23} & x_{33}\\end{matrix}\\right]$"
],
"text/plain": [
"[[x_11, x_12, x_13], [x_12, x_22, x_23], [x_13, x_23, x_33]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.create(name=\"x\", notation=\"voigt\")\n",
"display(A.data)\n",
"display(A.name)\n",
"display(A.notation)\n",
"display(A.to_general().data)"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & a c\\\\b & b d\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, a*c], [b, b*d]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([a, 0, 0, b], shape=(2, 2))\n",
"B = SymbolicTensor.from_list([1, c, 1, d], shape=(2, 2))\n",
"C = A @ B\n",
"display(C.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Converting"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Tensor to 3x3"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([a, b, c, b, d, e, c, e, f], shape=(3,3))\n",
"display(type(A))\n",
"B = A.to_3x3()\n",
"display(type(B))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### General to Symmetric 3x3"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"True"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicSymmetricThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([a, b, c, b, d, e, c, e, f], shape=(3,3))\n",
"display(type(A))\n",
"display(A.is_symmetric())\n",
"B = A.to_sym_3x3()\n",
"display(type(B))\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3x3 to Symmetric"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c\\\\b & d & e\\\\c & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, c], [b, d, e], [c, e, f]]"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"True"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicSymmetricThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicThreeByThreeTensor.from_list([a, b, c, b, d, e, c, e, f])\n",
"display(type(A))\n",
"display(A.data)\n",
"display(A.is_symmetric())\n",
"B = A.to_symmetric()\n",
"display(type(B))\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Converting to general"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicSymmetricThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c & d & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[a, b, c, d, e, f]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicSymmetricThreeByThreeTensor.from_list([a, b, c, d, e, f])\n",
"display(type(A))\n",
"display(A.data)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"mechpy.core.symbolic.tensor.SymbolicThreeByThreeTensor"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & c\\\\b & d & e\\\\c & e & f\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, c], [b, d, e], [c, e, f]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = A.to_general()\n",
"display(type(B))\n",
"display(B.data)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Multiplication"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Getting Tensor Components"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b\\\\c & d\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b], [c, d]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([a, b, c, d], shape=(2, 2))\n",
"display(A.data)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[[a, b], [c, d]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"display([[A[0, 0], A[0, 1]], [A[1, 0], A[1, 1]]])"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[[a, b], [c, d]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"display([[A[0][0], A[0][1]], [A[1][0], A[1][1]]])"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Eigenvalues and Eigenvectors"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(a,\n",
" 1,\n",
" [Matrix([\n",
" [1],\n",
" [0]])]),\n",
" (b,\n",
" 1,\n",
" [Matrix([\n",
" [0],\n",
" [1]])])]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([[a, 0], [0, b]], shape=(2, 2))\n",
"eigenvectors = A.eigenvectors()\n",
"display(eigenvectors)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[(a,\n",
" 2,\n",
" [Matrix([\n",
" [1],\n",
" [0],\n",
" [0]]),\n",
" Matrix([\n",
" [0],\n",
" [1],\n",
" [0]])]),\n",
" (b,\n",
" 1,\n",
" [Matrix([\n",
" [0],\n",
" [0],\n",
" [1]])])]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([[a, 0, 0], [0, a, 0], [0, 0, b]], shape=(3, 3))\n",
"eigenvectors = A.eigenvectors()\n",
"display(eigenvectors)"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{4}{17}\\\\- \\frac{8}{17}\\\\1\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[ 4/17],\n",
"[-8/17],\n",
"[ 1]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} - \\frac{\\sqrt{15} i}{6}\\\\1\\\\0\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[1/2 - sqrt(15)*I/6],\n",
"[ 1],\n",
"[ 0]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}\\frac{1}{2} + \\frac{\\sqrt{15} i}{6}\\\\1\\\\0\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[1/2 + sqrt(15)*I/6],\n",
"[ 1],\n",
"[ 0]])"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([[4, -2, 0], [3, 1, -4], [0, 0, 8]], shape=(3, 3))\n",
"eigenvectors = A.eigenvectors()\n",
"display(*[_[2][0] for _ in eigenvectors])"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle b$"
],
"text/plain": [
"b"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"1"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}0\\\\1\\\\0\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[0],\n",
"[1],\n",
"[0]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{a}{2} + \\frac{b}{2} - \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}$"
],
"text/plain": [
"a/2 + b/2 - sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"1"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}- \\frac{b}{a} + \\frac{\\frac{a}{2} + \\frac{b}{2} - \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}}{a}\\\\\\frac{- b + c}{c} + \\frac{b \\left(\\frac{a}{2} + \\frac{b}{2} - \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}\\right)}{a c}\\\\1\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[ -b/a + (a/2 + b/2 - sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2)/a],\n",
"[(-b + c)/c + b*(a/2 + b/2 - sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2)/(a*c)],\n",
"[ 1]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{a}{2} + \\frac{b}{2} + \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}$"
],
"text/plain": [
"a/2 + b/2 + sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"1"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}- \\frac{b}{a} + \\frac{\\frac{a}{2} + \\frac{b}{2} + \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}}{a}\\\\\\frac{- b + c}{c} + \\frac{b \\left(\\frac{a}{2} + \\frac{b}{2} + \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}\\right)}{a c}\\\\1\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[ -b/a + (a/2 + b/2 + sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2)/a],\n",
"[(-b + c)/c + b*(a/2 + b/2 + sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2)/(a*c)],\n",
"[ 1]])"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"A = SymbolicTensor.from_list([[a, 0, c], [a, b, b], [a, 0, b]], shape=(3, 3))\n",
"eigenvectors = A.eigenvectors()\n",
"for eig in eigenvectors:\n",
" eig_val = eig[0]\n",
" multiplicity = eig[1]\n",
" eig_vects = eig[2][0]\n",
" display(eig_val)\n",
" display(multiplicity)\n",
" display(eig_vects)"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}0 & \\frac{a - b - \\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2 a} & \\frac{a - b + \\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2 a}\\\\1 & \\frac{a \\left(- b + c\\right) + \\frac{b \\left(a + b - \\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}\\right)}{2}}{a c} & \\frac{a \\left(- b + c\\right) + \\frac{b \\left(a + b + \\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}\\right)}{2}}{a c}\\\\0 & 1 & 1\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[0, (a - b - sqrt(a**2 - 2*a*b + 4*a*c + b**2))/(2*a), (a - b + sqrt(a**2 - 2*a*b + 4*a*c + b**2))/(2*a)],\n",
"[1, (a*(-b + c) + b*(a + b - sqrt(a**2 - 2*a*b + 4*a*c + b**2))/2)/(a*c), (a*(-b + c) + b*(a + b + sqrt(a**2 - 2*a*b + 4*a*c + b**2))/2)/(a*c)],\n",
"[0, 1, 1]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}b & 0 & 0\\\\0 & \\frac{a}{2} + \\frac{b}{2} - \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2} & 0\\\\0 & 0 & \\frac{a}{2} + \\frac{b}{2} + \\frac{\\sqrt{a^{2} - 2 a b + 4 a c + b^{2}}}{2}\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[b, 0, 0],\n",
"[0, a/2 + b/2 - sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2, 0],\n",
"[0, 0, a/2 + b/2 + sqrt(a**2 - 2*a*b + 4*a*c + b**2)/2]])"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"P, D = A.diagonalize()\n",
"display(P)\n",
"display(D)"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}a & b & b\\\\b & 0 & c\\\\b & c & 0\\end{matrix}\\right]$"
],
"text/plain": [
"[[a, b, b], [b, 0, c], [b, c, 0]]"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"B = SymbolicSymmetricThreeByThreeTensor.from_list([a, b, b, 0, c, 0])\n",
"display(B.to_general().data)"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}0 & \\frac{a - c - \\sqrt{a^{2} - 2 a c + 8 b^{2} + c^{2}}}{2 b} & \\frac{a - c + \\sqrt{a^{2} - 2 a c + 8 b^{2} + c^{2}}}{2 b}\\\\-1 & 1 & 1\\\\1 & 1 & 1\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[ 0, (a - c - sqrt(a**2 - 2*a*c + 8*b**2 + c**2))/(2*b), (a - c + sqrt(a**2 - 2*a*c + 8*b**2 + c**2))/(2*b)],\n",
"[-1, 1, 1],\n",
"[ 1, 1, 1]])"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/latex": [
"$\\displaystyle \\left[\\begin{matrix}- c & 0 & 0\\\\0 & \\frac{a}{2} + \\frac{c}{2} - \\frac{\\sqrt{a^{2} - 2 a c + 8 b^{2} + c^{2}}}{2} & 0\\\\0 & 0 & \\frac{a}{2} + \\frac{c}{2} + \\frac{\\sqrt{a^{2} - 2 a c + 8 b^{2} + c^{2}}}{2}\\end{matrix}\\right]$"
],
"text/plain": [
"Matrix([\n",
"[-c, 0, 0],\n",
"[ 0, a/2 + c/2 - sqrt(a**2 - 2*a*c + 8*b**2 + c**2)/2, 0],\n",
"[ 0, 0, a/2 + c/2 + sqrt(a**2 - 2*a*c + 8*b**2 + c**2)/2]])"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"P, D = B.diagonalize()\n",
"display(P)\n",
"display(D)"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "venv",
"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": "3.10.12"
}
},
"nbformat": 4,
"nbformat_minor": 2
}