Revised list of TBCs

1 files changed, 65 insertions, 65 deletions

diff --git a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter6.ipynb b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter6.ipynb
index 9ad1cbe2..5499419a 100755..100644
--- a/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter6.ipynb
+++ b/Linear_Algebra_by_K._Hoffman_and_R._Kunze/Chapter6.ipynb
@@ -16,7 +16,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 26,
+   "execution_count": 1,
    "metadata": {
     "collapsed": false
    },
@@ -34,13 +34,13 @@
     }
    ],
    "source": [
-    "import sympy as sp\n",
+    "from sympy import Symbol,Matrix,eye\n",
     "print 'Standard ordered matrix for Linear operator T on R**2 is:'\n",
-    "A = sp.Matrix(([0, -1],[1 ,0]))\n",
+    "A = Matrix(([0, -1],[1 ,0]))\n",
     "print 'A = \\n',A\n",
     "print 'The characteristic polynomial for T or A is:',\n",
-    "x = sp.Symbol(\"x\")\n",
-    "p = (x*sp.eye(2)-A)\n",
+    "x = Symbol(\"x\")\n",
+    "p = (x*eye(2)-A)\n",
     "print p\n",
     "print 'Since this polynomial has no real roots,T has no characteristic values.'"
    ]
@@ -84,28 +84,28 @@
     }
    ],
    "source": [
-    "import sympy as sp\n",
-    "A = sp.Matrix(([3, 1, -1],[ 2, 2, -1],[2, 2, 0]))\n",
+    "from sympy import Symbol,Matrix,eye,solve\n",
+    "A = Matrix(([3, 1, -1],[ 2, 2, -1],[2, 2, 0]))\n",
     "print 'A = \\n',A\n",
     "print 'Characteristic polynomial for A is:',\n",
-    "x=sp.Symbol('x')\n",
+    "x=Symbol('x')\n",
     "p = A.charpoly(x)#\n",
     "print p.as_expr()\n",
     "print 'or'\n",
     "print '(x-1)(x-2)**2'\n",
     "\n",
-    "r = sp.solve(p.as_expr())#\n",
+    "r = solve(p.as_expr())#\n",
     "[m,n] = A.shape\n",
     "print 'The characteristic values of A are:'\n",
     "print r  #print round(r)\n",
-    "B = A-sp.eye(m)\n",
+    "B = A-eye(m)\n",
     "print 'Now, A-I = \\n',B\n",
     "\n",
     "print 'rank of A - I= ',B.rank()\n",
     "print 'So, nullity of T-I = 1'\n",
     "a1 = [1 ,0 ,2]#\n",
     "print 'The vector that spans the null space of T-I = ',a1\n",
-    "B = A-2*sp.eye(m)\n",
+    "B = A-2*eye(m)\n",
     "print 'Now,A-2I = \\n',B\n",
     "print 'rank of A - 2I= ',B.rank()\n",
     "print 'T*alpha = 2*alpha if alpha is a scalar multiple of a2'\n",
@@ -122,7 +122,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 28,
+   "execution_count": 1,
    "metadata": {
     "collapsed": false
    },
@@ -184,14 +184,14 @@
     }
    ],
    "source": [
-    "import sympy as sp\n",
-    "import numpy as np\n",
+    "from numpy import array,transpose,vstack,rank\n",
+    "from sympy import Symbol,Matrix,eye\n",
     "print 'Standard ordered matrix for Linear operator T on R**3 is:'\n",
-    "A = sp.Matrix(([5, -6, -6],[ -1, 4, 2],[ 3, -6, -4]))\n",
+    "A = Matrix(([5, -6, -6],[ -1, 4, 2],[ 3, -6, -4]))\n",
     "print 'A = \\n',A\n",
     "print 'xI - A = '\n",
-    "B = sp.eye(3)\n",
-    "x = sp.Symbol('x')\n",
+    "B = eye(3)\n",
+    "x = Symbol('x')\n",
     "P = x*B - A#\n",
     "print P\n",
     "\n",
@@ -211,7 +211,7 @@
     "print '=>'\n",
     "print ' * ', c\n",
     "print P\n",
-    "P = sp.Matrix(([P[0,0], P[0,2]],[P[2,0], P[2,2]]))\n",
+    "P = Matrix(([P[0,0], P[0,2]],[P[2,0], P[2,2]]))\n",
     "print '=>'\n",
     "print ' * ', c\n",
     "print P\n",
@@ -222,24 +222,24 @@
     "\n",
     "print 'Now, A - I = ',A-B\n",
     "print 'And, A- 2I = ',A-2*B\n",
-    "print 'rank(A-I) = ',np.rank(A-B)\n",
+    "print 'rank(A-I) = ',rank(A-B)\n",
     "\n",
-    "print 'rank(A-2I) = ',np.rank(A-2*B)\n",
+    "print 'rank(A-2I) = ',rank(A-2*B)\n",
     "print 'W1,W2 be the spaces of characteristic vectors associated with values 1,2'\n",
     "print 'So by theorem 2, T is diagonalizable'\n",
-    "a1 = np.array([[3, -1 ,3]])\n",
-    "a2 = np.array([[2, 1, 0]])\n",
-    "a3 = np.array([[2, 0, 1]])\n",
+    "a1 = array([[3, -1 ,3]])\n",
+    "a2 = array([[2, 1, 0]])\n",
+    "a3 = array([[2, 0, 1]])\n",
     "print 'Null space of (T- I) i.e basis of W1 is spanned by a1 = ',a1\n",
     "print 'Null space of (T- 2I) i.e. basis of W2 is spanned by vectors x1,x2,x3 such that x1 = 2x1 + 2x3'\n",
     "print 'One example :'\n",
     "print 'a2 = ',a2\n",
     "print 'a3 = ',a3\n",
     "print 'The diagonal matrix is:'\n",
-    "D = np.array([[1 ,0 ,0 ],[0, 2, 0],[0, 0, 2]])\n",
+    "D = array([[1 ,0 ,0 ],[0, 2, 0],[0, 0, 2]])\n",
     "print 'D = ',D\n",
     "print 'The standard basis matrix is denoted as:'\n",
-    "P = np.transpose(np.vstack([a1,a2,a3]))\n",
+    "P = transpose(vstack([a1,a2,a3]))\n",
     "print 'P = ',P\n",
     "print 'AP = ',A*P\n",
     "print 'PD = ',P*D\n",
@@ -256,7 +256,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 29,
+   "execution_count": 2,
    "metadata": {
     "collapsed": false
    },
@@ -298,37 +298,37 @@
     }
    ],
    "source": [
-    "import numpy as np\n",
-    "import sympy as sp\n",
+    "from numpy import array,transpose,vstack,rank\n",
+    "from sympy import Symbol,Matrix,eye\n",
     "\n",
-    "x = sp.Symbol(\"x\")\n",
-    "A = np.array([[5, -6, -6],[ -1, 4 ,2],[ 3, -6, -4]])   #Matrix given in Example 3\n",
+    "x = Symbol(\"x\")\n",
+    "A = array([[5, -6, -6],[ -1, 4 ,2],[ 3, -6, -4]])   #Matrix given in Example 3\n",
     "print 'A = \\n',A\n",
     "f = (x-1)*(x-2)**2# \n",
     "print 'Characteristic polynomial of A is:'\n",
     "print 'f = (x-1)(x-2)**2'\n",
     "print 'i.e., f = ',f\n",
     "p = (x-1)*(x-2)#\n",
-    "print '(A-I)(A-2I) = ',(A-sp.eye(3))*(A-2 * sp.eye(3))\n",
+    "print '(A-I)(A-2I) = ',(A-eye(3))*(A-2 * eye(3))\n",
     "print 'Since, (A-I)(A-2I) = 0. So, Minimal polynomial for above is:'\n",
     "print 'p = ',p\n",
     "print '---------------------------------------'\n",
     "\n",
-    "A = np.array([[3, 1 ,-1],[ 2, 2 ,-1],[2, 2, 0]])    #Matrix given in Example 2\n",
+    "A = array([[3, 1 ,-1],[ 2, 2 ,-1],[2, 2, 0]])    #Matrix given in Example 2\n",
     "print 'A = \\n',A\n",
     "f = (x-1)*(x-2)**2# \n",
     "print 'Characteristic polynomial of A is:'\n",
     "print 'f = (x-1)(x-2)**2'\n",
     "print 'i.e., f = ',f\n",
-    "print '(A-I)(A-2I) = ',(A-sp.eye(3))*(A-2 * sp.eye(3))\n",
+    "print '(A-I)(A-2I) = ',(A-eye(3))*(A-2 * eye(3))\n",
     "print 'Since, (A-I)(A-2I) is not equal to 0. T is not diagonalizable. So, Minimal polynomial cannot be p.'\n",
     "print '---------------------------------------'\n",
-    "A = np.array([[0, -1],[1, 0]])\n",
+    "A = array([[0, -1],[1, 0]])\n",
     "print 'A = \\n',A\n",
     "f = x**2 + 1#\n",
     "print 'Characteristic polynomial of A is:'\n",
     "print 'f = ',f\n",
-    "print 'A**2 + I = ',A**2 + sp.eye(2)\n",
+    "print 'A**2 + I = ',A**2 + eye(2)\n",
     "print 'Since, A**2 + I = 0, so minimal polynomial is'\n",
     "p = x**2 + 1\n",
     "print 'p = ',p"
@@ -343,7 +343,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 30,
+   "execution_count": 4,
    "metadata": {
     "collapsed": false
    },
@@ -352,7 +352,7 @@
      "name": "stdout",
      "output_type": "stream",
      "text": [
-      "A = \n",
+      " A = \n",
       "[[0 1 0 1]\n",
       " [1 0 1 0]\n",
       " [0 1 0 1]\n",
@@ -381,9 +381,9 @@
     }
    ],
    "source": [
-    "import numpy as np\n",
-    "import sympy as sp\n",
-    "A = np.array([[0, 1, 0, 1],[1, 0 ,1 ,0],[0, 1, 0, 1],[1, 0, 1, 0]])\n",
+    "from numpy import array,transpose,vstack,rank\n",
+    "from sympy import Symbol,Matrix,eye,solve\n",
+    "A = array([[0, 1, 0, 1],[1, 0 ,1 ,0],[0, 1, 0, 1],[1, 0, 1, 0]])\n",
     "print 'A = \\n',A\n",
     "print 'Computing powers on A:'\n",
     "print 'A**2 = \\n',A*A\n",
@@ -393,12 +393,12 @@
     "    return pp\n",
     "print 'if p = x**3 - 4x, then'\n",
     "print 'p(A) = ',p(A)\n",
-    "x = sp.Symbol(\"x\")\n",
+    "x = Symbol(\"x\")\n",
     "f = x**3 - 4*x\n",
     "print 'Minimal polynomial for A is: ',f\n",
-    "print 'Characteristic values for A are:',sp.solve(f,x)\n",
-    "print 'Rank(A) = ',np.rank(A)\n",
-    "print 'So, from theorem 2, characteristic polynomial for A is:',sp.Matrix(A).charpoly(x).as_expr()"
+    "print 'Characteristic values for A are:',solve(f,x)\n",
+    "print 'Rank(A) = ',rank(A)\n",
+    "print 'So, from theorem 2, characteristic polynomial for A is:',Matrix(A).charpoly(x).as_expr()"
    ]
   },
   {
@@ -410,7 +410,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 31,
+   "execution_count": 2,
    "metadata": {
     "collapsed": false
    },
@@ -420,51 +420,51 @@
      "output_type": "stream",
      "text": [
       "A = \n",
-      "[[  9.   1.   3.]\n",
-      " [ 10.   1.   3.]\n",
-      " [ 10.   5.   1.]]\n",
+      "[[ 9.  3.  3.]\n",
+      " [ 7.  4.  4.]\n",
+      " [ 1.  1.  2.]]\n",
       "A transpose is:\n",
       "A' = \n",
-      "[[  9.  10.  10.]\n",
-      " [  1.   1.   5.]\n",
-      " [  3.   3.   1.]]\n",
+      "[[ 9.  7.  1.]\n",
+      " [ 3.  4.  1.]\n",
+      " [ 3.  4.  2.]]\n",
       "Since, A' is not equal to A, A is not a symmetric matrix.\n",
       "Since, A' is not equal to -A, A is not a skew-symmetric matrix.\n",
       "A can be expressed as sum of A1 and A2\n",
       "i.e., A = A1 + A2\n",
       "A1 = \n",
-      "[[ 9.   5.5  6.5]\n",
-      " [ 5.5  1.   4. ]\n",
-      " [ 6.5  4.   1. ]]\n",
+      "[[ 9.   5.   2. ]\n",
+      " [ 5.   4.   2.5]\n",
+      " [ 2.   2.5  2. ]]\n",
       "A2 = \n",
-      "[[ 0.  -4.5 -3.5]\n",
-      " [ 4.5  0.  -1. ]\n",
-      " [ 3.5  1.   0. ]]\n",
+      "[[ 0.  -2.   1. ]\n",
+      " [ 2.   0.   1.5]\n",
+      " [-1.  -1.5  0. ]]\n",
       "A1 + A2 = \n",
-      "[[  9.   1.   3.]\n",
-      " [ 10.   1.   3.]\n",
-      " [ 10.   5.   1.]]\n"
+      "[[ 9.  3.  3.]\n",
+      " [ 7.  4.  4.]\n",
+      " [ 1.  1.  2.]]\n"
      ]
     }
    ],
    "source": [
-    "import numpy as np\n",
+    "from numpy import array,transpose,random,equal\n",
     "\n",
-    "A = np.random.rand(3,3)\n",
+    "A = random.rand(3,3)\n",
     "for i in range(0,3):\n",
     "    for j in range(0,3):\n",
     "        A[i,j]=round(A[i,j]*10)\n",
     "    \n",
     "print 'A = \\n',A\n",
     "print 'A transpose is:\\n',\n",
-    "Adash=np.transpose(A)\n",
+    "Adash=transpose(A)\n",
     "print \"A' = \\n\",Adash\n",
-    "if np.equal(Adash,A).all():\n",
+    "if equal(Adash,A).all():\n",
     "    print \"Since, A' = A, A is a symmetric matrix.\"\n",
     "else:\n",
     "    print \"Since, A' is not equal to A, A is not a symmetric matrix.\"\n",
     "\n",
-    "if np.equal(Adash,-A).all():\n",
+    "if equal(Adash,-A).all():\n",
     "    print \"Since, A' = -A, A is a skew-symmetric matrix.\"\n",
     "else:\n",
     "    print \"Since, A' is not equal to -A, A is not a skew-symmetric matrix.\"\n",