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"3.6\n",
"\n",
"To prove ||.|| is a norm, we have to show that ||.|| satisfies the following three conditions:\n",
"\n",
"1) $ ||x|| \\geq 0 $ and $ ||x|| = 0 $ only if $ x = 0 $ \n",
"\n",
"2) $ ||x+y|| \\leq ||x|| + ||y|| $ (Triangle Inequality)\n",
"\n",
"3) $ || {\\alpha} x || = ||\\alpha||\\cdot ||x|| $\n",
"\n",
"Proof:\n",
"\n",
"1) $||y|| =1 $ so only if $ x= 0 $, $||x||' = 0$ and $ ||x||' $ is a kind of norm so $ ||x||' \\geq 0 $ \n",
"\n",
"\n",
"2) $||x+z||'= \\sup_{||y||=1}{| y^*(x+z)|}= \\sup_{||y||=1}{| y^*x + y^*z|} \\leq \\sup_{||y||=1}{|y^*x|} + \\sup_{||y||=1}{|y^*z|} = ||x||' + ||z||' $ Therefore,\n",
"\n",
"$||x+z||' \\leq ||x||'+||z||' $ (Triangle Inequality is satisfied.)\n",
"\n",
"3) For any scalars $\\alpha \\in C $, $ ||\\alpha x ||'=\\sup_{||y||=1}{|y^*(\\alpha x)|}= \\sup_{||y||=1}{|\\alpha| \\cdot |y^*x|} = |\\alpha| \\cdot \\sup_{||y||=1}{|y^*x|} = |\\alpha| \\cdot ||x||' $\n",
"\n",
"4.5\n",
"\n",
"$A*A$ is real and symmetric. It means that it has a complete set of orthongonal eigenvectors : $v_1,v_2,v_3,...v_n$. $V=[v_1,v_2,v_3,...,v_n] $ and $\\lambda_i$ is the corresponding eigenvalue to $v_i$. We assume $ \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3 \\geq ... \\geq \\lambda_n$. If the rank of A is $r$ then $\\lambda_{r+1} = \\lambda_{r+2}= ... = \\lambda_{n} = 0 $\n",
"\n",
"So $null(A^*A)=null(A)$.\n",
"\n",
"Let us define $ \\sigma_i = \\sqrt{\\lambda_i} $ and $u_i = \\frac{Av_i}{\\sigma_i} $ where $(i=1,2,...,r)$. Then $ {u_1,u_2,...,u_r} $ is an orthogonal set of vectors.\n",
"\n",
"$u_i^* u_j = \\frac{(Av_i)^* (Av_j)}{\\sigma_i \\sigma_j} = \\frac{ v_i^*(A^* A)v_j }{\\sqrt{\\lambda_i \\lambda_j}} = \\frac{ \\lambda_j v_i^* v_j }{ \\sqrt {\\lambda_i \\lambda_j } } = \\delta_{ij} $\n",
"\n",
"$U^*AV = [u_1,u_2,...,u_m]^* A [v_1,v_2,...,v_n] = (u_i^* A v_j)$\n",
"\n",
"$ (u_i^* A v_j) = $\n",
"$ \\begin {cases}\n",
" 0 & j > r \\\\\n",
" \\sigma_i \\delta_{ij} & j\\leq r\n",
" \\end {cases} $\n",
" \n",
" It shows that $ U^* A V $ is a diagonal matrix. So A has a real SVD.\n"
]
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"source": [
"function Svd = SVD (A)\n",
" \n",
" ATA=A.'*A;\n",
" Trace = ATA(1,1)+ATA(2,2);\n",
" Determinant= det(ATA);\n",
" \n",
" L1= (1/2)*(Trace+ sqrt((Trace^2)-(4*Determinant)));\n",
" L2= (1/2)*(Trace- sqrt((Trace^2)-(4*Determinant)));\n",
" \n",
" SingVal1=sqrt(L1);\n",
" SingVal2=sqrt(L2);\n",
" \n",
" %To be able to write singular values in a descending order\n",
" if(SingVal1<SingVal2) \n",
" Temp=SingVal1;\n",
" SingVal1=SingVal2;\n",
" SingVal2=Temp;\n",
" endif\n",
" \n",
" \n",
" %To find eigenvectors associated with eigenvalues\n",
" \n",
" %First eigenvector\n",
" DeltaMatrix1=[(ATA(1,1)-L1),ATA(1,2);ATA(2,1),(ATA(2,2)-L1)];\n",
" \n",
" PV1=[-(DeltaMatrix1(1,2));(DeltaMatrix1(1,1))];\n",
" \n",
" Abs1=abs(PV1(1,1));\n",
" Abs2=abs(PV1(2,1));\n",
" Range=-1;\n",
" \n",
" if(Abs1<Abs2)\n",
" Range=Abs2;\n",
" else\n",
" Range=Abs1;\n",
" endif \n",
" \n",
" X=PV1(1,1);\n",
" Y=PV1(2,1);\n",
" \n",
" Divisor1=1;\n",
" for i= 2:Range\n",
" if(mod(int16(X),i)==0 && mod(int16(Y),i)==0)\n",
" Divisor1=i;\n",
" endif\n",
" endfor \n",
" \n",
" \n",
" V1=[PV1(1,1)/Divisor1;PV1(2,1)/Divisor1];\n",
" \n",
" Length1= sqrt(V1(1,1)^2+V1(2,1)^2);\n",
" \n",
" %First Eigenvector is VV1\n",
" VV1=[V1(1,1)/Length1;V1(2,1)/Length1];\n",
" \n",
" \n",
" %Finding Second Eigenvector\n",
" \n",
" DeltaMatrix2=[(ATA(1,1)-L2),ATA(1,2);ATA(2,1),(ATA(2,2)-L2)];\n",
" \n",
" PV2=[-(DeltaMatrix2(1,2));(DeltaMatrix2(1,1))];\n",
" \n",
" Abs1=abs(PV2(1,1));\n",
" Abs2=abs(PV2(2,1));\n",
" Range=-1;\n",
" \n",
" if(Abs1<Abs2)\n",
" Range=Abs2;\n",
" else\n",
" Range=Abs1;\n",
" endif \n",
" \n",
" X=PV2(1,1);\n",
" Y=PV2(2,1);\n",
" \n",
" Divisor2=1;\n",
" for i= 2:Range\n",
" if(mod(int16(X),i)==0 && mod(int16(Y),i)==0)\n",
" Divisor2=i;\n",
" endif\n",
" endfor \n",
" \n",
" V2=[PV2(1,1)/Divisor2;PV2(2,1)/Divisor2];\n",
" \n",
" Length2= sqrt(V2(1,1)^2+V2(2,1)^2);\n",
" \n",
" %Second Eigenvector is VV2\n",
" VV2=[V2(1,1)/Length2;V2(2,1)/Length2];\n",
" \n",
" \n",
" \n",
" %Building V Matrix. V matrix is eigenvectors of A.'*A.\n",
" VV1=VV1.';\n",
" VV2=VV2.';\n",
" V=[VV1;VV2];\n",
" \n",
" disp(\"Part a\");\n",
" V=V.'\n",
" \n",
" % A=U.SigmaMatrix.V* ===> U=A.V.(SigmaMatrix)^-1\n",
" \n",
" SigmaMatrix=[SingVal1,0;0,SingVal2];\n",
" InverseSigmaMatrix=inv(SigmaMatrix);\n",
" U=A*V*InverseSigmaMatrix\n",
" \n",
" %Svd=U*SigmaMatrix*V.'\n",
" \n",
" \n",
" \n",
" disp(\"Part c\")\n",
" \n",
" FirstRowSum= A(1,1)+A(1,2);\n",
" SecondRowSum= A(2,1)+A(2,2);\n",
" \n",
" FirstColumnSum= A(1,1)+A(2,1);\n",
" SecondColumnSum= A(1,2)+A(2,2);\n",
" \n",
" %To find 1-norm of A by Example 3.3 on textbook\n",
" \n",
" if(FirstColumnSum > SecondColumnSum)\n",
" MaxColumnSum=FirstColumnSum;\n",
" else\n",
" MaxColumnSum=SecondColumnSum;\n",
" endif\n",
" \n",
" %To find infinity norm of A by Example 3.4 on textbook\n",
" if(FirstRowSum>SecondRowSum)\n",
" MaxRowSum=FirstRowSum;\n",
" else\n",
" MaxRowSum=SecondRowSum;\n",
" endif\n",
" \n",
" \n",
" disp(\"The value of 1-norm of A equals to maximum column sum by Example 3.3 on textbook: \"),disp(MaxColumnSum)\n",
" disp(\"The value of 2-norm of A is maximum singular value : \"), disp(SingVal1)\n",
" disp(\"The value of Frobenius norm of A is square root of sum of squares of singular values: \"),disp(sqrt((SingVal1)^2+(SingVal2)^2))\n",
" disp(\"The value of infinity norm of A equals to maximum row sum by Example 3.4 on textbook: \"),disp(MaxRowSum)\n",
" disp(\"\")\n",
" disp(\"Part d\")\n",
" \n",
" disp(\"A=U*SigmaMatrix*V.' so Inv(A)= Inv(V.')*Inv(SigmaMatrix)*Inv(U)=V*Inv(SigmaMatrix)*U.'\")\n",
" disp(\"\")\n",
" InverseOfA=V*inv(SigmaMatrix)*U.'\n",
" \n",
" \n",
" disp(\"Part e\")\n",
" \n",
" Trace = A(1,1)+A(2,2);\n",
" Determinant= det(A);\n",
" \n",
" FirstEigenValueOfA= (1/2)*(Trace+ sqrt((Trace^2)-(4*Determinant)))\n",
" SecondEigenValueOfA= (1/2)*(Trace- sqrt((Trace^2)-(4*Determinant)))\n",
" disp(\"\")\n",
" \n",
" disp(\"Part f\")\n",
" \n",
" Mult1=FirstEigenValueOfA*SecondEigenValueOfA;\n",
" Mult2=SingVal1*SingVal2;\n",
" disp(\"Multiplication of the eigenvalues of A is \"),disp(int16(Mult1))\n",
" disp(\"Multiplication of the singular values of A is \"),disp(int16(Mult2))\n",
" disp(\"Determinant of A is\"),disp(det(A))\n",
" \n",
" disp(\"\")\n",
" disp(\"Part g\")\n",
" disp(\"The area of ellipsoid is multiplication of singular values so\"),disp(int16(Mult2))\n",
" \n",
" disp(\"Part b\");\n",
" \n",
" t=0:0.1:(2*pi);\n",
" \n",
" %Unit circle\n",
" x1=cos(t);\n",
" y1=sin(t);\n",
" \n",
" %Ellipse\n",
" \n",
" w=A*[x1;y1];\n",
" x2= w(1,:);\n",
" y2= w(2,:);\n",
" \n",
" %Right singular vectors\n",
" \n",
" v1=V(:,1);\n",
" v2=V(:,2);\n",
" \n",
" %Left singular vectors\n",
" \n",
" u1=U(:,1);\n",
" u2=U(:,2);\n",
" \n",
" %Scaled left singular vectors\n",
" \n",
" w1=SingVal1*u1;\n",
" w2=SingVal2*u2;\n",
" \n",
" %Plotting the unit cirlce and the right singular vectors\n",
" \n",
" figure(1);\n",
" axis([-1.2 1.2 -1.2 1.2]);\n",
" plot(x1,y1);\n",
" hold on;\n",
" plotv(v1, 'b');\n",
" hold on\n",
" plotv(v2, 'r');\n",
" grid on;\n",
" \n",
" %Plotting the ellipse and the scaled left singular vectors\n",
" figure(2);\n",
" axis([-3 3 -3 3]);\n",
" plot(x2,y2);\n",
" hold on;\n",
" plotv(w1,'b');\n",
" hold on;\n",
" plotv(w2,'r');\n",
" grid on;\n",
" \n",
" \n",
"endfunction"
]
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"cell_type": "code",
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"name": "stdout",
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"text": [
"Part a\n",
"V =\n",
"\n",
" 0.60000 0.80000\n",
" -0.80000 0.60000\n",
"\n",
"U =\n",
"\n",
" -0.70711 0.70711\n",
" -0.70711 -0.70711\n",
"\n",
"Part c\n",
"The value of 1-norm of A equals to maximum column sum by Example 3.3 on textbook: \n",
" 16\n",
"The value of 2-norm of A is maximum singular value : \n",
" 14.142\n",
"The value of Frobenius norm of A is square root of sum of squares of singular values: \n",
" 15.811\n",
"The value of infinity norm of A equals to maximum row sum by Example 3.4 on textbook: \n",
" 9\n",
"\n",
"Part d\n",
"A=U*SigmaMatrix*V.' so Inv(A)= Inv(V.')*Inv(SigmaMatrix)*Inv(U)=V*Inv(SigmaMatrix)*U.'\n",
"\n",
"InverseOfA =\n",
"\n",
" 0.050000 -0.110000\n",
" 0.100000 -0.020000\n",
"\n",
"Part e\n",
"FirstEigenValueOfA = 1.5000 + 9.8869i\n",
"SecondEigenValueOfA = 1.5000 - 9.8869i\n",
"\n",
"Part f\n",
"Multiplication of the eigenvalues of A is \n",
"100\n",
"Multiplication of the singular values of A is \n",
"100\n",
"Determinant of A is\n",
" 100\n",
"\n",
"Part g\n",
"The area of ellipsoid is multiplication of singular values so\n",
"100\n",
"Part b\n",
"error: 'plotv' undefined near line 202 column 5\n",
"error: called from:\n",
"error: SVD at line 202, column 5\n"
]
}
],
"source": [
"SVD([-2,11;-10,5])"
]
}
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