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synced cells with lecture notes
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‎Tutorial_06b_MatrixAlgebra.ipynb

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@@ -184,6 +184,98 @@
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"printmat(A')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Vectors: Inner and Outer Products\n",
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"\n",
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"There are several different ways to think about a vector in applied mathematics: as a $K \\times 1$ matrix (a column vector), a $1 \\times K$ matrix (a row vector) or just a flat $K$ vector. Julia uses flat vectors but they are mostly interchangable with column vectors. \n",
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"\n",
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"The inner product of two (column) vectors with $K$ elements is calculated as `x'z` or `dot(x,y)` (textbook: $x'z$ or $x \\cdot z$) to get a scalar. (The dot is obtained by `\\cdot + TAB`, but this is sometimes hard to distinguish from or things like `x.z`.)\n",
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"\n",
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"In contrast, the outer product of two (column) vectors with $K$ elements is calculated as `x*z'` (textbook: $xz'$) to get a $K\\times K$ matrix."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\u001b[34m\u001b[1mx and z\u001b[22m\u001b[39m\n",
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" 10 2 \n",
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" 11 5 \n",
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"\n",
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"\u001b[34m\u001b[1mx'z: \u001b[22m\u001b[39m\n",
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" 75 \n",
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"\u001b[34m\u001b[1mx*z':\u001b[22m\u001b[39m\n",
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" 20 50 \n",
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" 22 55 \n",
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"\n"
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]
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}
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],
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"source": [
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"x = [10,11] #[10;11] gives the same\n",
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"z = [2,5]\n",
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"printblue(\"x and z\")\n",
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"printmat([x z])\n",
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"\n",
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"printblue(\"x'z: \")\n",
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"printlnPs(x'z) #dot(x,z) gives the same\n",
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"\n",
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"printblue(\"x*z':\")\n",
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"printmat(x*z')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Vectors: Quadratic Forms\n",
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"\n",
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"A quadratic form ($A$ is an $n \\times n$ matrix and $x$ is an $n$ vector): `x'A*x` (textbook: $x'Ax$) to get a scalar. There is also the form `dot(x,A,x)`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\u001b[34m\u001b[1mx:\u001b[22m\u001b[39m\n",
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" 10 \n",
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" 11 \n",
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"\n",
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"\u001b[34m\u001b[1mA:\u001b[22m\u001b[39m\n",
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" 1 3 \n",
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" 3 4 \n",
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"\n",
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"\u001b[34m\u001b[1mx'A*x: \u001b[22m\u001b[39m\n",
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" 1244 \n"
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]
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}
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],
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"source": [
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"A = [1 3;3 4]\n",
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"x = [10,11]\n",
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"\n",
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"printblue(\"x:\")\n",
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"printmat(x)\n",
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"printblue(\"A:\")\n",
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"printmat(A)\n",
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"\n",
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"printblue(\"x'A*x: \")\n",
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"printlnPs(x'A*x) #or dot(x,A,x)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
@@ -206,7 +298,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"execution_count": 7,
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"metadata": {
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"tags": []
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},
@@ -257,7 +349,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"execution_count": 8,
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"metadata": {},
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"outputs": [
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{
@@ -302,7 +394,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
@@ -340,7 +432,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
@@ -375,98 +467,6 @@
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"printmat(1I[1:3,2])"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Vectors: Inner and Outer Products\n",
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"\n",
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"There are several different ways to think about a vector in applied mathematics: as a $K \\times 1$ matrix (a column vector), a $1 \\times K$ matrix (a row vector) or just a flat $K$ vector. Julia uses flat vectors but they are mostly interchangable with column vectors. \n",
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"\n",
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"The inner product of two (column) vectors with $K$ elements is calculated as `x'z` or `dot(x,y)` (textbook: $x'z$ or $x \\cdot z$) to get a scalar. (The dot is obtained by `\\cdot + TAB`, but this is sometimes hard to distinguish from or things like `x.z`.)\n",
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"\n",
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"In contrast, the outer product of two (column) vectors with $K$ elements is calculated as `x*z'` (textbook: $xz'$) to get a $K\\times K$ matrix."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\u001b[34m\u001b[1mx and z\u001b[22m\u001b[39m\n",
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" 10 2 \n",
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" 11 5 \n",
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"\n",
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"\u001b[34m\u001b[1mx'z: \u001b[22m\u001b[39m\n",
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" 75 \n",
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"\u001b[34m\u001b[1mx*z':\u001b[22m\u001b[39m\n",
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" 20 50 \n",
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" 22 55 \n",
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"\n"
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]
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}
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],
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"source": [
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"x = [10,11] #[10;11] gives the same\n",
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"z = [2,5]\n",
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"printblue(\"x and z\")\n",
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"printmat([x z])\n",
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"\n",
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"printblue(\"x'z: \")\n",
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"printlnPs(x'z) #dot(x,z) gives the same\n",
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"\n",
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"printblue(\"x*z':\")\n",
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"printmat(x*z')"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Vectors: Quadratic Forms\n",
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"\n",
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"A quadratic form ($A$ is an $n \\times n$ matrix and $x$ is an $n$ vector): `x'A*x` (textbook: $x'Ax$) to get a scalar. There is also the form `dot(x,A,x)`."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"\u001b[34m\u001b[1mx:\u001b[22m\u001b[39m\n",
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" 10 \n",
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" 11 \n",
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"\n",
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"\u001b[34m\u001b[1mA:\u001b[22m\u001b[39m\n",
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" 1 3 \n",
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" 3 4 \n",
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"\n",
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"\u001b[34m\u001b[1mx'A*x: \u001b[22m\u001b[39m\n",
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" 1244 \n"
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]
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}
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],
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"source": [
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"A = [1 3;3 4]\n",
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"x = [10,11]\n",
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"\n",
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"printblue(\"x:\")\n",
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"printmat(x)\n",
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"printblue(\"A:\")\n",
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"printmat(A)\n",
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"\n",
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"printblue(\"x'A*x: \")\n",
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"printlnPs(x'A*x) #or dot(x,A,x)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"A[1,:]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"1×2 Matrix{Int64}:\n",
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" 1 3"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"A[1:1,:]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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},
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"anaconda-cloud": {},
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"kernelspec": {
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"display_name": "Julia 1.11.1",
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"display_name": "Julia 1.11.3",
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"language": "julia",
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"name": "julia-1.11"
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},
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.11.1"
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"version": "1.11.3"
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}
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},
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"nbformat": 4,

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