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 {
		   "cell_type": "markdown",
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	   "source": [
       "# Chapter 7: Linear Algebra"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.1: Orthonomal_basis.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
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"source": [
"        //    PG (470)\n",
"u1 = [1/2,sqrt(3)/2]\n",
"u2 = [-sqrt(3)/2,1/2]\n",
"//    For a given vector x = (x1,x2), it can be written as\n",
"//                    x = alpha1*u1 + alpha2*u2\n",
"//    alpha1 = (x1+x2*sqrt(3))/2\n",
"//    alpha2 = (x2-x1*sqrt(3))/2\n",
"//    (1,0) = (1/2)*u1 - (sqrt(3)/2)*u2"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.2: Canonical_forms.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"        //    PG (476)\n",
"A = [0.2 0.6 0;1.6 -0.2 0;-1.6 1.2 3.0]\n",
"U = [0.6 0 -0.8;0.8 0 0.6;0 1.0 0]\n",
"Ustar = inv(U)\n",
"T = Ustar*A*U\n",
"trace(A)\n",
"lam =spec(A)'\n",
"lam1 = lam(1,1)\n",
"lam2 = lam(1,2)\n",
"lam3 = lam(1,3)\n",
"lam1 + lam2 + lam3\n",
"    //    trace(A) = lam1 + lam2 + lam3\n",
"det(A)\n",
"lam1*lam2*lam3\n",
"    //    det(A) = lam1 * lam2 * lam3"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.3: Orthonomal_eigen_vectors.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"        //    PG (477)\n",
"A = [2 1 0;1 3 1;0 1 2]\n",
"lam = spec(A)'\n",
"lam1 = lam(1,1)\n",
"lam2 = lam(1,2)\n",
"lam3 = lam(1,3)\n",
"//    Orthonomal Eigen vectors\n",
"u1 = (1/sqrt(3))*[1;-1;1]\n",
"u2 = (1/sqrt(2))*[1;0;-1]\n",
"u3 = (1/sqrt(6))*[1;2;1]"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.4: Vector_and_matrix_norms.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"        //    PG (481)\n",
"x = [1,0,-1,2]\n",
"    //    1-norm\n",
"norm(x,1)\n",
"    //    2-norm\n",
"norm(x,2)\n",
"    //    infinity norm\n",
"norm(x,'inf')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.5: Frobenious_norm.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"        //    PG (484)\n",
"//    A be n * n \n",
"//    norm(A*x,2)\n",
"//    norm(A*x,2) <= norm(A,'fro') * norm(x,2)\n",
"//    norm(A*B,'fro') = norm(A,'fro') * norm(B,'fro')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.6: Norm.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"        //    PG (489)\n",
"A = [1 -2;-3 4]\n",
"norm(A,1)\n",
"norm(A,2)\n",
"norm(A,'inf')\n",
"lam = spec(A)\n",
"r = max(abs(lam))\n",
"    //r < = norm(A,2)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 7.7: Inverse_exists.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"        //    PG (494)\n",
"A = [4 1 0 0;1 4 1 0;0 1 4 1;0 0 1 4]\n",
"B = A/4 - eye()\n",
"norm(B,'inf')\n",
"    //    Let (I+B = C)\n",
"C = eye() + B\n",
"inv(C)\n",
"//    Inverse of (I + B) exists\n",
"norm(C,'inf')\n",
"//    Inverse of A exists."
   ]
   }
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		   "help_links": [
			{
			 "text": "MetaKernel Magics",
			 "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
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