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ME3255S2017/lecture_09/lecture_09.ipynb

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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Linear Algebra (Review/Introduction)\n",
"\n",
"Representation of linear equations:\n",
"\n",
"1. $5x_{1}+3x_{2} =1$\n",
"\n",
"2. $x_{1}+2x_{2}+3x_{3} =2$\n",
"\n",
"3. $x_{1}+x_{2}+x_{3} =3$\n",
"\n",
"in matrix form:\n",
"\n",
"$\\left[ \\begin{array}{ccc}\n",
"5 & 3 & 0 \\\\\n",
"1 & 2 & 3 \\\\\n",
"1 & 1 & 1 \\end{array} \\right]\n",
"\\left[\\begin{array}{c} \n",
"x_{1} \\\\ \n",
"x_{2} \\\\\n",
"x_{3}\\end{array}\\right]=\\left[\\begin{array}{c} \n",
"1 \\\\\n",
"2 \\\\\n",
"3\\end{array}\\right]$\n",
"\n",
"$Ax=b$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Vectors \n",
"\n",
"column vector x (length of 3):\n",
"\n",
"$\\left[\\begin{array}{c} \n",
"x_{1} \\\\ \n",
"x_{2} \\\\\n",
"x_{3}\\end{array}\\right]$\n",
"\n",
"row vector y (length of 3):\n",
"\n",
"$\\left[\\begin{array}{ccc} \n",
"y_{1}~ \n",
"y_{2}~ \n",
"y_{3}\\end{array}\\right]$\n",
"\n",
"vector of length N:\n",
"\n",
"$\\left[\\begin{array}{c} \n",
"x_{1} \\\\ \n",
"x_{2} \\\\\n",
"\\vdots \\\\\n",
"x_{N}\\end{array}\\right]$\n",
"\n",
"The $i^{th}$ element of x is $x_{i}$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"x =\n",
"\n",
" 1 2 3 4 5 6 7 8 9 10\n",
"\n"
]
}
],
"source": [
"x=[1:10]"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans =\n",
"\n",
" 1\n",
" 2\n",
" 3\n",
" 4\n",
" 5\n",
" 6\n",
" 7\n",
" 8\n",
" 9\n",
" 10\n",
"\n"
]
}
],
"source": [
"x'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Matrices\n",
"\n",
"Matrix A is 3x3:\n",
"\n",
"$A=\\left[ \\begin{array}{ccc}\n",
"5 & 3 & 0 \\\\\n",
"1 & 2 & 3 \\\\\n",
"1 & 1 & 1 \\end{array} \\right]$\n",
"\n",
"elements in the matrix are denoted $A_{row~column}$, $A_{23}=3$\n",
"\n",
"In general, matrix, B, can be any size, $M \\times N$ (M-rows and N-columns):\n",
"\n",
"$B=\\left[ \\begin{array}{cccc}\n",
"B_{11} & B_{12} &...& B_{1N} \\\\\n",
"B_{21} & B_{22} &...& B_{2N} \\\\\n",
"\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
"B_{M1} & B_{M2} &...& B_{MN}\\end{array} \\right]$"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 5 3 0\n",
" 1 2 3\n",
" 1 1 1\n",
"\n"
]
}
],
"source": [
"A=[5,3,0;1,2,3;1,1,1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Multiplication\n",
"\n",
"A column vector is a $1\\times N$ matrix and a row vector is a $M\\times 1$ matrix\n",
"\n",
"**Multiplying matrices is not commutative**\n",
"\n",
"$A B \\neq B A$\n",
"\n",
"Inner dimensions must agree, output is outer dimensions. \n",
"\n",
"A is $M1 \\times N1$ and B is $M2 \\times N2$\n",
"\n",
"C=AB\n",
"\n",
"Therefore N1=M2 and C is $M1 \\times N2$\n",
"\n",
"If $C'=BA$, then N2=M1 and C is $M2 \\times N1$\n",
"\n",
"\n",
"e.g. \n",
"$A=\\left[ \\begin{array}{cc}\n",
"5 & 3 & 0 \\\\\n",
"1 & 2 & 3 \\end{array} \\right]$\n",
"\n",
"$B=\\left[ \\begin{array}{cccc}\n",
"1 & 2 & 3 & 4 \\\\\n",
"5 & 6 & 7 & 8 \\\\\n",
"9 & 10 & 11 & 12 \\end{array} \\right]$\n",
"\n",
"C=AB\n",
"\n",
"$[2\\times 4] = [2 \\times 3][3 \\times 4]$\n",
"\n",
"The rule for multiplying matrices, A and B, is\n",
"\n",
"$C_{ij} = \\sum_{k=1}^{n}A_{ik}B_{kj}$\n",
"\n",
"In the previous example, \n",
"\n",
"$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$\n",
"\n",
"\n",
"Multiplication is associative:\n",
"\n",
"$(AB)C = A(BC)$\n",
"\n",
"and distributive:\n",
"\n",
"$A(B+C)=AB+AC$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 5 3 0\n",
" 1 2 3\n",
"\n",
"B =\n",
"\n",
" 1 2 3 4\n",
" 5 6 7 8\n",
" 9 10 11 12\n",
"\n"
]
}
],
"source": [
"A=[5,3,0;1,2,3] \n",
"B=[1,2,3,4;5,6,7,8;9,10,11,12]"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"C =\n",
"\n",
" 0 0 0 0\n",
" 0 0 0 0\n",
"\n",
"C =\n",
"\n",
" 20 28 36 44\n",
" 38 44 50 56\n",
"\n"
]
}
],
"source": [
"C=zeros(2,4)\n",
"C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);\n",
"C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);\n",
"\n",
"C=A*B"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)\r\n"
]
}
],
"source": [
"Cp=B*A"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Representation of a problem with Matrices and Vectors\n",
"\n",
"If you have a set of known output, $y_{1},~y_{2},~...y_{N}$ and a set of equations that\n",
"relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, then these can be written in a vector\n",
"matrix format as:\n",
"\n",
"$y=Ax=\\left[\\begin{array}{c} \n",
"y_{1} \\\\ \n",
"y_{2} \\\\\n",
"\\vdots \\\\\n",
"y_{N}\\end{array}\\right]\n",
"=\n",
"A\\left[\\begin{array}{c} \n",
"x_{1} \\\\ \n",
"x_{2} \\\\\n",
"\\vdots \\\\\n",
"x_{N}\\end{array}\\right]$\n",
"\n",
"$A=\n",
"\\left[\\begin{array}{cccc} \n",
"| & | & & | \\\\ \n",
"a_{1} & a_{2} & ... & a_{N} \\\\ \n",
"| & | & & | \\end{array}\\right]$\n",
"\n",
"or \n",
"\n",
"$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$\n",
"\n",
"where each $a_{i}$ is a column vector and $x_{i}$ is a scalar taken from the $i^{th}$\n",
"element of x.\n",
"\n",
"Consider the following problem, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n",
"\n",
"![Springs-masses](mass_springs.svg)\n",
"\n",
"The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n",
"\n",
"$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n",
"\n",
"$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n",
"\n",
"$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n",
"\n",
"$m_{4}g-k(x_{4}-x_{3})=0$\n",
"\n",
"in matrix form:\n",
"\n",
"$\\left[ \\begin{array}{cccc}\n",
"2k & -k & 0 & 0 \\\\\n",
"-k & 2k & -k & 0 \\\\\n",
"0 & -k & 2k & -k \\\\\n",
"0 & 0 & -k & k \\end{array} \\right]\n",
"\\left[ \\begin{array}{c}\n",
"x_{1} \\\\\n",
"x_{2} \\\\\n",
"x_{3} \\\\\n",
"x_{4} \\end{array} \\right]=\n",
"\\left[ \\begin{array}{c}\n",
"m_{1}g \\\\\n",
"m_{2}g \\\\\n",
"m_{3}g \\\\\n",
"m_{4}g \\end{array} \\right]$"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"K =\n",
"\n",
" 20 -10 0 0\n",
" -10 20 -10 0\n",
" 0 -10 20 -10\n",
" 0 0 -10 10\n",
"\n",
"y =\n",
"\n",
" 9.8100\n",
" 19.6200\n",
" 29.4300\n",
" 39.2400\n",
"\n",
"x =\n",
"\n",
" 9.8100\n",
" 18.6390\n",
" 25.5060\n",
" 29.4300\n",
"\n"
]
}
],
"source": [
"k=10; % N/m\n",
"m1=1; % kg\n",
"m2=2;\n",
"m3=3;\n",
"m4=4;\n",
"g=9.81; % m/s^2\n",
"K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n",
"y=[m1*g;m2*g;m3*g;m4*g]\n",
"\n",
"x=K\\y"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Matrix Operations\n",
"\n",
"Identity matrix `eye(M,N)` **Assume M=N unless specfied**\n",
"\n",
"$I_{ij} =1$ if $i=j$ \n",
"\n",
"$I_{ij} =0$ if $i\\neq j$ \n",
"\n",
"AI=A=IA\n",
"\n",
"Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any\n",
"number\n",
"\n",
"$D_{ij} =d_{i}$ if $i=j$ \n",
"\n",
"$D_{ij} =0$ if $i\\neq j$ "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Transpose\n",
"\n",
"The transpose of a matrix changes the rows -> columns and columns-> rows\n",
"\n",
"$(A^{T})_{ij} = A_{ji}$\n",
"\n",
"for diagonal matrices, $D^{T}=D$\n",
"\n",
"in general, the transpose has the following qualities:\n",
" \n",
"1. $(A^{T})^{T}=A$\n",
"\n",
"2. $(AB)^{T} = B^{T}A^{T}$\n",
"\n",
"3. $(A+B)^{T} = A^{T} +B^{T}$ \n",
"\n",
"All matrices have a symmetric and antisymmetric part:\n",
"\n",
"$A= 1/2(A+A^{T}) +1/2(A-A^{T})$\n",
"\n",
"If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric."
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans =\n",
"\n",
" 20 38\n",
" 28 44\n",
" 36 50\n",
" 44 56\n",
"\n",
"ans =\n",
"\n",
" 20 38\n",
" 28 44\n",
" 36 50\n",
" 44 56\n",
"\n",
"error: operator *: nonconformant arguments (op1 is 3x2, op2 is 4x3)\n"
]
}
],
"source": [
"(A*B)'\n",
"\n",
"B'*A' % if this wasnt true, then inner dimensions wouldn;t match\n",
"\n",
"A'*B' == (A*B)'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Square matrix operations\n",
"\n",
"### Trace\n",
"\n",
"The trace of a square matrix is the sum of the diagonal elements\n",
"\n",
"$tr~A=\\sum_{i=1}^{N}A_{ii}$\n",
"\n",
"The trace is invariant, meaning that if you change the basis through a rotation, then the\n",
"trace remains the same. \n",
"\n",
"It also has the following properties:\n",
"\n",
"1. $tr~A=tr~A^{T}$\n",
"\n",
"2. $tr~A+tr~B=tr(A+B)$\n",
"\n",
"3. $tr(kA)=k tr~A$\n",
"\n",
"4. $tr(AB)=tr(BA)$"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"id_m =\n",
"\n",
"Diagonal Matrix\n",
"\n",
" 1 0 0\n",
" 0 1 0\n",
" 0 0 1\n",
"\n",
"ans = 3\n"
]
}
],
"source": [
"id_m=eye(3)\n",
"trace(id_m)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Inverse\n",
"\n",
"The inverse of a square matrix, $A^{-1}$ is defined such that\n",
"\n",
"$A^{-1}A=I=AA^{-1}$\n",
"\n",
"Not all square matrices have an inverse, they can be *singular* or *non-invertible*\n",
"\n",
"The inverse has the following properties:\n",
"\n",
"1. $(A^{-1})^{-1}=A$\n",
"\n",
"2. $(AB)^{-1}=B^{-1}A^{-1}$\n",
"\n",
"3. $(A^{-1})^{T}=(A^{T})^{-1}$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 0.5762106 0.3533174 0.7172134\n",
" 0.7061664 0.4863733 0.9423436\n",
" 0.4255961 0.0016613 0.3561407\n",
"\n"
]
}
],
"source": [
"A=rand(3,3)"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Ainv =\n",
"\n",
" 41.5613 -30.1783 -3.8467\n",
" 36.2130 -24.2201 -8.8415\n",
" -49.8356 36.1767 7.4460\n",
"\n"
]
}
],
"source": [
"Ainv=inv(A)"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"B =\n",
"\n",
" 0.524529 0.470856 0.708116\n",
" 0.084491 0.730986 0.528292\n",
" 0.303545 0.782522 0.389282\n",
"\n"
]
}
],
"source": [
"B=rand(3,3)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans =\n",
"\n",
" -182.185 125.738 40.598\n",
" -133.512 97.116 17.079\n",
" 282.422 -200.333 -46.861\n",
"\n",
"ans =\n",
"\n",
" -182.185 125.738 40.598\n",
" -133.512 97.116 17.079\n",
" 282.422 -200.333 -46.861\n",
"\n",
"ans =\n",
"\n",
" 41.5613 36.2130 -49.8356\n",
" -30.1783 -24.2201 36.1767\n",
" -3.8467 -8.8415 7.4460\n",
"\n",
"ans =\n",
"\n",
" 41.5613 36.2130 -49.8356\n",
" -30.1783 -24.2201 36.1767\n",
" -3.8467 -8.8415 7.4460\n",
"\n"
]
}
],
"source": [
"inv(A*B)\n",
"inv(B)*inv(A)\n",
"\n",
"inv(A')\n",
"\n",
"inv(A)'"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Orthogonal Matrices\n",
"\n",
"Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A\n",
"square matrix is *orthogonal* if all its column vectors are orthogonal to each other and\n",
"normalized. The column vectors are then called *orthonormal* and the following results\n",
"\n",
"$U^{T}U=I=UU^{T}$\n",
"\n",
"and \n",
"\n",
"$||Ux||_{2}=||x||_{2}$\n",
"\n",
"### Determinant\n",
"\n",
"The **determinant** of a matrix has 3 properties\n",
"\n",
"1. The determinant of the identity matrix is one, $|I|=1$\n",
"\n",
"2. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:\n",
"\n",
"$t|A|=\\left[ \\begin{array}{cccc}\n",
"tA_{11} & tA_{12} &...& tA_{1N} \\\\\n",
"A_{21} & A_{22} &...& A_{2N} \\\\\n",
"\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
"A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n",
"\n",
"3. If you switch 2 rows, the determinant changes sign:\n",
"\n",
"\n",
"$-|A|=\\left[ \\begin{array}{cccc}\n",
"A_{21} & A_{22} &...& A_{2N} \\\\\n",
"A_{11} & A_{12} &...& A_{1N} \\\\\n",
"\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
"A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n",
"\n",
"4. inverse of the determinant is the determinant of the inverse:\n",
"\n",
"$|A^{-1}|=\\frac{1}{|A|}=|A|^{-1}$\n",
"\n",
"For a $2\\times2$ matrix, \n",
"\n",
"$|A|=\\left|\\left[ \\begin{array}{cc}\n",
"A_{11} & A_{12} \\\\\n",
"A_{21} & A_{22} \\\\\n",
"\\end{array} \\right]\\right| = A_{11}A_{22}-A_{21}A_{12}$\n",
"\n",
"For a $3\\times3$ matrix,\n",
"\n",
"$|A|=\\left|\\left[ \\begin{array}{ccc}\n",
"A_{11} & A_{12} & A_{13} \\\\\n",
"A_{21} & A_{22} & A_{23} \\\\\n",
"A_{31} & A_{32} & A_{33}\\end{array} \\right]\\right|=$\n",
"\n",
"$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}\n",
"-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$\n",
"\n",
"For larger matrices, the determinant is \n",
"\n",
"$|A|=\n",
"\\frac{1}{n!}\n",
"\\sum_{i_{r}}^{N}\n",
"\\sum_{j_{r}}^{N}\n",
"\\epsilon_{i_{1}...i_{N}}\n",
"\\epsilon_{j_{1}...j_{N}}\n",
"A_{i_{1}j_{1}}\n",
"A_{i_{2}j_{2}}\n",
"...\n",
"A_{i_{N}j_{N}}$\n",
"\n",
"Where the Levi-Cevita symbol is $\\epsilon_{i_{1}i_{2}...i_{N}} = 1$ if it is an even\n",
"permutation and $\\epsilon_{i_{1}i_{2}...i_{N}} = -1$ if it is an odd permutation and\n",
"$\\epsilon_{i_{1}i_{2}...i_{N}} = 0$ if $i_{n}=i_{m}$ e.g. $i_{1}=i_{7}$\n",
"\n",
"![Levi-Cevita rule for even and odd permutations for an $N\\times N$\n",
"determinant](even_odd_perm.svg)\n",
"\n",
"So a $4\\times4$ matrix determinant,\n",
"\n",
"$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
"A_{11} & A_{12} & A_{13} & A_{14} \\\\\n",
"A_{21} & A_{22} & A_{23} & A_{24} \\\\\n",
"A_{31} & A_{32} & A_{33} & A_{34} \\\\\n",
"A_{41} & A_{42} & A_{43} & A_{44} \\end{array} \\right]\\right|$\n",
"\n",
"$=\\epsilon_{1234}\\epsilon_{1234}A_{11}A_{22}A_{33}A_{44}+\n",
"\\epsilon_{2341}\\epsilon_{1234}A_{21}A_{32}A_{43}A_{14}+\n",
"\\epsilon_{3412}\\epsilon_{1234}A_{31}A_{42}A_{13}A_{24}+\n",
"\\epsilon_{4123}\\epsilon_{1234}A_{41}A_{12}A_{23}A_{34}+...\n",
"\\epsilon_{4321}\\epsilon_{4321}A_{44}A_{33}A_{22}A_{11}$\n",
"\n",
"### Special Case for determinants\n",
"\n",
"The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. \n",
"\n",
"Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j<i$), the\n",
"determinant is \n",
"\n",
"$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
"A_{11} & A_{12} &...& A_{1N} \\\\\n",
"0 & A_{22} &...& A_{2N} \\\\\n",
"0 & 0 &\\ddots & \\vdots \\\\\n",
"0 & 0 &...& A_{NN}\\end{array} \\right]\\right|=A_{11}A_{22}A_{33}...A_{NN}$"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"D =\n",
"\n",
" 1 0 0 0\n",
" 0 2 0 0\n",
" 0 0 3 0\n",
" 0 0 0 4\n",
"\n",
"A =\n",
"\n",
" 1 2 3 4\n",
" 0 2 3 4\n",
" 0 0 3 4\n",
" 0 0 0 4\n",
"\n"
]
}
],
"source": [
"D=[1,0,0,0;0,2,0,0;0,0,3,0;0,0,0,4]\n",
"A=[1,2,3,4;0,2,3,4;0,0,3,4;0,0,0,4] % upper triangular matrix (4x4)"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 24\n",
"ans = 24\n"
]
}
],
"source": [
"det(D)\n",
"det(A)"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"A =\n",
"\n",
" 2 4 6 8\n",
" 0 2 3 4\n",
" 0 0 3 4\n",
" 0 0 0 4\n",
"\n"
]
}
],
"source": [
"A(1,:)=2*A(1,:)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 48\r\n"
]
}
],
"source": [
"det(A)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 0.020833\r\n"
]
}
],
"source": [
"det(inv(A))"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = 0.020833\r\n"
]
}
],
"source": [
"1/48"
]
},
{
"cell_type": "code",
"execution_count": 39,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"r1 =\n",
"\n",
" 2 4 6 8\n",
"\n",
"r2 =\n",
"\n",
" 0 2 3 4\n",
"\n",
"A =\n",
"\n",
" 0 2 3 4\n",
" 0 2 3 4\n",
" 0 0 3 4\n",
" 0 0 0 4\n",
"\n",
"A =\n",
"\n",
" 0 2 3 4\n",
" 2 4 6 8\n",
" 0 0 3 4\n",
" 0 0 0 4\n",
"\n"
]
}
],
"source": [
"r1=A(1,:)\n",
"r2=A(2,:)\n",
"A(1,:)=r2\n",
"A(2,:)=r1"
]
},
{
"cell_type": "code",
"execution_count": 40,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans = -48\r\n"
]
}
],
"source": [
"det(A)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Working with matrices (or arrays)\n",
"\n",
"When you need a row of your matrix, use `A(1,:)`\n",
"\n",
"When you need a column of your matrix, use `A(:,1)`"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"B =\n",
"\n",
" Columns 1 through 6:\n",
"\n",
" 9.3678e-02 6.6083e-01 8.6163e-01 2.4581e-01 3.2483e-01 6.9693e-01\n",
" 4.5401e-01 5.6867e-01 1.4516e-01 9.5440e-02 6.8525e-01 6.1453e-03\n",
" 5.4096e-01 9.6443e-01 8.2454e-02 1.8644e-01 7.0369e-01 2.9403e-03\n",
" 8.3509e-01 1.1006e-01 5.8880e-01 4.9634e-01 9.3148e-02 8.3110e-01\n",
" 5.9499e-01 8.8513e-01 6.5640e-01 5.3244e-01 2.9278e-01 7.6347e-02\n",
" 1.5456e-01 3.1275e-01 4.5873e-01 7.2297e-01 3.6996e-01 1.2478e-01\n",
" 4.7960e-01 1.8491e-01 6.1816e-01 3.3450e-01 5.9307e-01 8.9796e-01\n",
" 9.2872e-01 3.8687e-01 8.3991e-01 8.8064e-01 6.0058e-01 3.5300e-01\n",
" 7.3063e-01 2.0543e-01 7.0693e-01 5.0973e-01 7.1647e-01 1.3979e-01\n",
" 2.3365e-01 3.5988e-01 4.8453e-01 9.6969e-01 9.8916e-01 2.7608e-02\n",
"\n",
" Columns 7 and 8:\n",
"\n",
" 8.8934e-01 1.7184e-01\n",
" 2.5948e-03 7.2273e-01\n",
" 1.1076e-01 9.3209e-01\n",
" 8.3082e-01 3.6414e-01\n",
" 9.6406e-01 2.4431e-01\n",
" 2.6633e-04 9.9584e-01\n",
" 4.4733e-01 4.2158e-01\n",
" 1.2252e-01 8.7513e-01\n",
" 5.2620e-01 8.7209e-01\n",
" 5.8721e-02 9.7262e-01\n",
"\n"
]
}
],
"source": [
"B=rand(10,8)"
]
},
{
"cell_type": "code",
"execution_count": 44,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans =\n",
"\n",
" 0.861626\n",
" 0.145156\n",
" 0.082454\n",
" 0.588799\n",
" 0.656403\n",
" 0.458729\n",
" 0.618158\n",
" 0.839911\n",
" 0.706933\n",
" 0.484532\n",
"\n"
]
}
],
"source": [
"B(:,3) % print 3 column of matrix B"
]
},
{
"cell_type": "code",
"execution_count": 45,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"ans =\n",
"\n",
" Columns 1 through 7:\n",
"\n",
" 0.594987 0.885135 0.656403 0.532440 0.292784 0.076347 0.964064\n",
"\n",
" Column 8:\n",
"\n",
" 0.244310\n",
"\n"
]
}
],
"source": [
"B(5,:)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"How to determine, determinant numerically?\n",
"'' , inverse of matrices\n",
"'' , solutions , what is x_1, x_2, ..."
]
}
],
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"text": "MetaKernel Magics",
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