A=[1 2 3;4 5 6;7 8 10] A = 1 2 3 4 5 6 7 8 10 rank(A) ans = 3 [Q,R]=qr(A) Q = -0.1231 0.9045 0.4082 -0.4924 0.3015 -0.8165 -0.8616 -0.3015 0.4082 R = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 Q'*Q-eye(3) ans = 1.0e-15 * 0.4441 -0.5265 -0.2552 -0.5265 0.6661 0.3378 -0.2552 0.3378 0.2220 eig(A) ans = 16.7075 -0.9057 0.1982 A A = 1 2 3 4 5 6 7 8 10 x=A(:,1) x = 1 4 7 v=x+[norm(x);0;0] v = 9.1240 4.0000 7.0000 H1=eye(3)-2/(v'*v)*v*v' H1 = -0.1231 -0.4924 -0.8616 -0.4924 0.7841 -0.3777 -0.8616 -0.3777 0.3389 H1*x ans = -8.1240 0.0000 0.0000 ans(2:3) ans = 1.0e-15 * 0.3331 0.3331 A1=H1*A A1 = -8.1240 -9.6011 -11.9399 0.0000 -0.0860 -0.5497 0.0000 -0.9004 -1.4619 A1([2,3],1)=0 A1 = -8.1240 -9.6011 -11.9399 0 -0.0860 -0.5497 0 -0.9004 -1.4619 x=A1([2,3],2) x = -0.0860 -0.9004 v=x-[norm(x);0] v = -0.9905 -0.9004 HH2=eye(2)-2/(v'*v)*v*v' HH2 = -0.0950 -0.9955 -0.9955 0.0950 HH2*x ans = 0.9045 0.0000 H2=eye(3); H2([2,3],[2,3])=HH2 H2 = 1.0000 0 0 0 -0.0950 -0.9955 0 -0.9955 0.0950 A2=H2*A1 A2 = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0.0000 0.4082 A2(3,2) ans = 1.2490e-16 A2(3,2)=0 A2 = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 R R = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 Q1=H1*H2 Q1 = -0.1231 0.9045 0.4082 -0.4924 0.3015 -0.8165 -0.8616 -0.3015 0.4082 Q Q = -0.1231 0.9045 0.4082 -0.4924 0.3015 -0.8165 -0.8616 -0.3015 0.4082 A3=A2; A3(1,1)=-A3(1,1) A3 = 8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 R1=A2; Q1*R1 ans = 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 10.0000 Q*R ans = 1.0000 2.0000 3.0000 4.0000 5.0000 6.0000 7.0000 8.0000 10.0000 R2=A3 R2 = 8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 Q2=A/R2 Q2 = 0.1231 3.5176 -2.0412 0.4924 10.7539 -10.6145 0.8616 17.9902 -16.7382 Q Q = -0.1231 0.9045 0.4082 -0.4924 0.3015 -0.8165 -0.8616 -0.3015 0.4082 R R = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 Q2=Q; Q2 Q2 = -0.1231 0.9045 0.4082 -0.4924 0.3015 -0.8165 -0.8616 -0.3015 0.4082 Q2(:,1)=-Q2(:,1) Q2 = 0.1231 0.9045 0.4082 0.4924 0.3015 -0.8165 0.8616 -0.3015 0.4082 Q2'*Q2-eye(3) ans = 1.0e-15 * 0.4441 0.5265 0.2552 0.5265 0.6661 0.3378 0.2552 0.3378 0.2220 R2=Q2'*A R2 = 8.1240 9.6011 11.9399 0.0000 0.9045 1.5076 0.0000 0.0000 0.4082 R R = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 Q,R Q = -0.1231 0.9045 0.4082 -0.4924 0.3015 -0.8165 -0.8616 -0.3015 0.4082 R = -8.1240 -9.6011 -11.9399 0 0.9045 1.5076 0 0 0.4082 Q2.R2 {Dot indexing is not supported for variables of this type.} Q2,R2 Q2 = 0.1231 0.9045 0.4082 0.4924 0.3015 -0.8165 0.8616 -0.3015 0.4082 R2 = 8.1240 9.6011 11.9399 0.0000 0.9045 1.5076 0.0000 0.0000 0.4082 diary off