A
A =
1 2 3
4 5 6
7 8 10
x=A([1 2],1)
x =
1
4
r=norm(x)
r =
4.1231
c=x(1)/r
c =
0.2425
s=-x(2)/r
s =
-0.9701
G=[c -s;s c]
G =
0.2425 0.9701
-0.9701 0.2425
G'*G-eye(2)
ans =
1.0e-17 *
0 0.5317
0.5317 0
G12=eye(3);
G12([1 2],[1 2])=G
G12 =
0.2425 0.9701 0
-0.9701 0.2425 0
0 0 1.0000
A1=G12*A
A1 =
4.1231 5.3358 6.5485
0 -0.7276 -1.4552
7.0000 8.0000 10.0000
x=A1([1 3],1)
x =
4.1231
7.0000
r=norm(x);
c=x(1)/r
c =
0.5075
s=-x(2)/r
s =
-0.8616
G=[c -s;s c]
G =
0.5075 0.8616
-0.8616 0.5075
G'*G-eye(2)
ans =
1.0e-15 *
-0.1110 -0.0062
-0.0062 -0.1110
G13=eye(3);
G13([1,3],[1 3])=G
G13 =
0.5075 0 0.8616
0 1.0000 0
-0.8616 0 0.5075
A2=G13*A1
A2 =
8.1240 9.6011 11.9399
0 -0.7276 -1.4552
-0.0000 -0.5374 -0.5672
A2(3,1)
ans =
-4.4409e-16
A2(3,1)=0;
x=A2([2 3],2)
x =
-0.7276
-0.5374
r=norm(x);
c=x(1)/r;
s=-x(2)/r;
G=[c -s;s c]
G =
-0.8044 -0.5941
0.5941 -0.8044
G'*G-eye(2)
ans =
1.0e-16 *
0 -0.1102
-0.1102 0
G23=eye(3);
G23([2 3],[2 3])=G
G23 =
1.0000 0 0
0 -0.8044 -0.5941
0 0.5941 -0.8044
A3=G23*A2
A3 =
8.1240 9.6011 11.9399
0 0.9045 1.5076
0 0.0000 -0.4082
A3(3,2)
ans =
5.5511e-17
A3(3,2)=0;
R=A3
R =
8.1240 9.6011 11.9399
0 0.9045 1.5076
0 0 -0.4082
G=G23*G13*G12
G =
0.1231 0.4924 0.8616
0.9045 0.3015 -0.3015
-0.4082 0.8165 -0.4082
G'*G-eye(3)
ans =
1.0e-15 *
0 -0.0245 -0.0331
-0.0245 -0.1110 -0.0607
-0.0331 -0.0607 -0.2220
Q=G'
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
[Q1,R1]=qr(A)
Q1 =
-0.1231 0.9045 0.4082
-0.4924 0.3015 -0.8165
-0.8616 -0.3015 0.4082
R1 =
-8.1240 -9.6011 -11.9399
0 0.9045 1.5076
0 0 0.4082
A=[1 2 3;4 5 6;7 8 9]
A =
1 2 3
4 5 6
7 8 9
[Q1,R1]=qr(A)
Q1 =
-0.1231 0.9045 0.4082
-0.4924 0.3015 -0.8165
-0.8616 -0.3015 0.4082
R1 =
-8.1240 -9.6011 -11.0782
0 0.9045 1.8091
0 0 -0.0000
R1(3,3)
ans =
-1.1116e-15
x=A([1 2],1)
x =
1
4
r=norm(x);c=x(1)/r;s=-x(2)/r;
G=[c -s;s c];
G12=eye(3);
G12([1 2],[1 2])=G;
A1=G12*A
A1 =
4.1231 5.3358 6.5485
0 -0.7276 -1.4552
7.0000 8.0000 9.0000
x=A1([1 3],1);
r=norm(x);c=x(1)/r;s=-x(2)/r;
G=[c -s;s c];
G13=eye(3);
G13([1,3],[1 3])=G;
A2=G13*A1
A2 =
8.1240 9.6011 11.0782
0 -0.7276 -1.4552
-0.0000 -0.5374 -1.0747
A2(3,1)=0;
x=A2([2 3],2);
r=norm(x);c=x(1)/r;s=-x(2)/r;
G=[c -s;s c];
G23=eye(3);
G23([2 3],[2 3])=G
G23 =
1.0000 0 0
0 -0.8044 -0.5941
0 0.5941 -0.8044
A3=G23*A2
A3 =
8.1240 9.6011 11.0782
0 0.9045 1.8091
0 0.0000 0
A3(3,2)=0
A3 =
8.1240 9.6011 11.0782
0 0.9045 1.8091
0 0 0
R=A3;
G=G23*G13*G12;
Q=G'
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 11.0782
0 0.9045 1.8091
0 0 0
[Q1,R1]=qr(A)
Q1 =
-0.1231 0.9045 0.4082
-0.4924 0.3015 -0.8165
-0.8616 -0.3015 0.4082
R1 =
-8.1240 -9.6011 -11.0782
0 0.9045 1.8091
0 0 -0.0000
det(R)
ans =
0
det(R1)
ans =
8.1689e-15
det(A)
ans =
-9.5162e-16
A=[1 1 1 1 1;1 2 3 4 5]'
A =
1 1
1 2
1 3
1 4
1 5
[Q,R]=qr(A)
Q =
-0.4472 -0.6325 -0.4149 -0.3626 -0.3104
-0.4472 -0.3162 0.0672 0.3996 0.7320
-0.4472 0.0000 0.8377 -0.2013 -0.2403
-0.4472 0.3162 -0.2174 0.6543 -0.4739
-0.4472 0.6325 -0.2726 -0.4900 0.2925
R =
-2.2361 -6.7082
0 3.1623
0 0
0 0
0 0
A=[1 1 1 1 1;1 2 3 4 5]
A =
1 1 1 1 1
1 2 3 4 5
[Q,R]=qr(A)
Q =
-0.7071 -0.7071
-0.7071 0.7071
R =
-1.4142 -2.1213 -2.8284 -3.5355 -4.2426
0 0.7071 1.4142 2.1213 2.8284
Q*R-A
ans =
1.0e-14 *
-0.0333 -0.0555 -0.1332 -0.0666 -0.0888
0 -0.0222 -0.0444 0 0
A=[1 1 1 1 1;1 1 3 4 5]
A =
1 1 1 1 1
1 1 3 4 5
A=[1 1 1 1 1;1 1 3 4 5]'
A =
1 1
1 1
1 3
1 4
1 5
[Q,R]=qr(A)
Q =
-0.4472 -0.5031 -0.4384 -0.4269 -0.4153
-0.4472 -0.5031 0.0759 0.3590 0.6421
-0.4472 0.0559 0.8314 -0.2089 -0.2492
-0.4472 0.3354 -0.2125 0.6893 -0.4089
-0.4472 0.6149 -0.2563 -0.4125 0.4313
R =
-2.2361 -6.2610
0 3.5777
0 0
0 0
0 0
A=[1 1 1 1 1;2 2 2 2 2]'
A =
1 2
1 2
1 2
1 2
1 2
[Q,R]=qr(A)
Q =
-0.4472 0.8944 -0.0000 -0.0000 -0.0000
-0.4472 -0.2236 -0.5000 -0.5000 -0.5000
-0.4472 -0.2236 0.8333 -0.1667 -0.1667
-0.4472 -0.2236 -0.1667 0.8333 -0.1667
-0.4472 -0.2236 -0.1667 -0.1667 0.8333
R =
-2.2361 -4.4721
0 -0.0000
0 0
0 0
0 0
R(2,2)
ans =
-2.3269e-16
A=[1 1 1 1 1;1 1 1 1 1]'
A =
1 1
1 1
1 1
1 1
1 1
R(2,2)
ans =
-2.3269e-16
[Q,R]=qr(A)
Q =
-0.4472 0.8944 -0.0000 -0.0000 -0.0000
-0.4472 -0.2236 -0.5000 -0.5000 -0.5000
-0.4472 -0.2236 0.8333 -0.1667 -0.1667
-0.4472 -0.2236 -0.1667 0.8333 -0.1667
-0.4472 -0.2236 -0.1667 -0.1667 0.8333
R =
-2.2361 -2.2361
0 -0.0000
0 0
0 0
0 0
R(2,2)
ans =
-1.1634e-16
A
A =
1 1
1 1
1 1
1 1
1 1
U=[1+i 0; 0 (1-i)/2]
U =
1.0000 + 1.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.5000 - 0.5000i
U'*U
ans =
2.0000 0
0 0.5000
U=[(1+i)/sqrt(2) 0; 0 (1-i)/sqrt(2)]
U =
0.7071 + 0.7071i 0.0000 + 0.0000i
0.0000 + 0.0000i 0.7071 - 0.7071i
U'*U
ans =
1.0000 0
0 1.0000
det(U)
ans =
1.0000
U=[i 0;0 1]
U =
0.0000 + 1.0000i 0.0000 + 0.0000i
0.0000 + 0.0000i 1.0000 + 0.0000i
U'*U
ans =
1 0
0 1
det(U)
ans =
0.0000 + 1.0000i
det(U')
ans =
0.0000 - 1.0000i
U=0.5*[1+i1-i;1-i 1+i]
{Undefined function or variable 'i1'.}
U=0.5*[1+i 1-i;1-i 1+i]
U =
0.5000 + 0.5000i 0.5000 - 0.5000i
0.5000 - 0.5000i 0.5000 + 0.5000i
U'*U
ans =
1 0
0 1
det(U)
ans =
0.0000 + 1.0000i
det(U')
ans =
0.0000 - 1.0000i
A=[1 2 3;4 5 6;7 8 9]
A =
1 2 3
4 5 6
7 8 9
[Q,R]=qr(A);
A=R*Q
A =
15.2727 -6.9031 0.0000
-2.0041 -0.2727 -0.0000
0.0000 0.0000 -0.0000
[Q,R]=qr(A);A=R*Q
A =
16.1586 -4.7469 -0.0000
0.1520 -1.1586 0.0000
0.0000 -0.0000 -0.0000
[Q,R]=qr(A);A=R*Q
A =
16.1139 -4.9095 0.0000
-0.0105 -1.1139 -0.0000
0.0000 0.0000 -0.0000
[Q,R]=qr(A);A=R*Q
A =
16.1171 -4.8983 -0.0000
0.0007 -1.1171 0.0000
0.0000 -0.0000 -0.0000
[Q,R]=qr(A);A=R*Q
A =
16.1168 -4.8990 0.0000
-0.0001 -1.1168 -0.0000
0.0000 0.0000 -0.0000
[Q,R]=qr(A);A=R*Q
A =
16.1168 -4.8990 -0.0000
0.0000 -1.1168 0.0000
0.0000 -0.0000 -0.0000
format long
A
A =
16.116844961530521 -4.898975996867128 -0.000000000000005
0.000003488699230 -1.116844961530516 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843901083847 -4.898979727321645 0.000000000000005
-0.000000241755286 -1.116843901083842 -0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843974569342 -4.898979468813518 -0.000000000000005
0.000000016752842 -1.116843974569337 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969477042 -4.898979486727276 0.000000000000005
-0.000000001160917 -1.116843969477035 -0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969829922 -4.898979485485912 -0.000000000000005
0.000000000080448 -1.116843969829915 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969805469 -4.898979485571934 0.000000000000005
-0.000000000005575 -1.116843969805462 -0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807164 -4.898979485565973 -0.000000000000005
0.000000000000386 -1.116843969807156 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807046 -4.898979485566386 0.000000000000005
-0.000000000000027 -1.116843969807039 -0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807054 -4.898979485566358 -0.000000000000005
0.000000000000002 -1.116843969807047 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807054 -4.898979485566359 0.000000000000005
-0.000000000000000 -1.116843969807046 -0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807054 -4.898979485566359 -0.000000000000005
0.000000000000000 -1.116843969807046 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807054 -4.898979485566359 0.000000000000005
-0.000000000000000 -1.116843969807046 -0.000000000000000
0.000000000000000 0.000000000000000 -0.000000000000000
[Q,R]=qr(A);A=R*Q
A =
16.116843969807054 -4.898979485566359 -0.000000000000005
0.000000000000000 -1.116843969807046 0.000000000000000
0.000000000000000 -0.000000000000000 -0.000000000000000
A(3,:)
ans =
1.0e-15 *
0.000000000000000 -0.000000000000000 -0.453828149038388
A(2,1)
ans =
4.277731343481284e-20
eig([1 2 3;4 5 6;7 8 9])
ans =
16.116843969807043
-1.116843969807042
-0.000000000000001
A=[1 2 3;4 5 6;7 8 10]
A =
1 2 3
4 5 6
7 8 10
B=A;
[Q,R]=qr(A);A=R*Q
A =
16.015151515151523 -6.643313090275599 -0.351763235340715
-1.744333604709241 -0.181818181818182 -0.123091490979333
-0.351763235340724 -0.123091490979332 0.166666666666667
[Q,R]=qr(A);A=R*Q
A =
16.734901091206172 -4.591852870124714 -1.387640361961922
0.103994577180329 -0.924995774381694 -0.163293978368585
0.004329289496538 -0.056348940814890 0.190094683175528
[Q,R]=qr(A);A=R*Q
A =
16.705972325154956 -4.778103139904833 1.101530965515702
-0.005618047608844 -0.905188921702173 0.087249784220137
-0.000051633457236 -0.012258112213257 0.199216596547222
[Q,R]=qr(A);A=R*Q
A =
16.707575645373918 -4.756835669230492 -1.166034089869882
0.000304694911433 -0.905573495906818 -0.102572861891548
0.000000612011625 -0.002679769739270 0.197997850532904
[Q,R]=qr(A);A=R*Q
A =
16.707488852274018 -4.760586669730308 1.151953726732982
-0.000016517107063 -0.905788488580077 0.099285175888949
-0.000000007263903 -0.000586825609606 0.198299636306062
[Q,R]=qr(A);A=R*Q
A =
16.707493558112098 -4.759821952928859 -1.155037578385545
0.000000895428649 -0.905728788028376 -0.100001571180982
0.000000000086187 -0.000128428785502 0.198235229916282
[Q,R]=qr(A);A=R*Q
A =
16.707493303006178 -4.759986628855633 1.154362647968516
-0.000000048542524 -0.905742708764995 0.099844969526661
-0.000000000001023 -0.000028110989789 0.198249405758822
[Q,R]=qr(A);A=R*Q
A =
16.707493316835937 -4.759950748132400 -1.154510379953173
0.000000002631570 -0.905739623575714 -0.099879236721606
0.000000000000012 -0.000006152850940 0.198246306739784
[Q,R]=qr(A);A=R*Q
A =
16.707493316086204 -4.759958593591898 1.154478044747155
-0.000000000142662 -0.905740301314316 0.099871736961665
-0.000000000000000 -0.000001346727159 0.198246985228119
[Q,R]=qr(A);A=R*Q
A =
16.707493316126847 -4.759956876865901 -1.154485122232361
0.000000000007734 -0.905740152857232 -0.099873378467943
0.000000000000000 -0.000000294769261 0.198246836730390
[Q,R]=qr(A);A=R*Q
A =
16.707493316124644 -4.759957252596051 1.154483573124729
-0.000000000000419 -0.905740185358372 0.099873019179551
-0.000000000000000 -0.000000064518596 0.198246869233733
[Q,R]=qr(A);A=R*Q
A =
16.707493316124765 -4.759957170358262 -1.154483912190787
0.000000000000023 -0.905740178244235 -0.099873097819896
0.000000000000000 -0.000000014121720 0.198246862119477
for k=1:10, [Q,R]=qr(A);A=R*Q; end
A
A =
16.707493316124758 -4.759957185125937 -1.154483851303510
0.000000000000000 -0.905740179521761 -0.099873083698179
0.000000000000000 -0.000000000000004 0.198246863397010
eig(B)
ans =
16.707493316124744
-0.905740179521759
0.198246863397009
help hess
hess Hessenberg form.
H = hess(A) is the Hessenberg form of the matrix A.
The Hessenberg form of a matrix is zero below the first
subdiagonal and has the same eigenvalues as A. If the matrix
is symmetric or Hermitian, the form is tridiagonal.
[P,H] = hess(A) produces a unitary matrix P and a Hessenberg
matrix H so that A = P*H*P' and P'*P = EYE(SIZE(P)).
[AA,BB,Q,Z] = hess(A,B) for square matrices A and B, produces
an upper Hessenberg matrix AA, an upper triangular matrix BB,
and unitary matrices Q and Z such that
Q*A*Z = AA, Q*B*Z = BB.
Reference page for hess
A=B;
A=hess(A)
A =
1.000000000000000 -3.597007303087045 -0.248069469178417
-8.062257748298549 14.799999999999997 2.399999999999999
0 0.399999999999997 0.199999999999999
B
B =
1 2 3
4 5 6
7 8 10
for k=1:10, [Q,R]=qr(A);A=R*Q; end
A
A =
16.707493316126847 4.759957131908719 -1.154484070687116
-0.000000000007734 -0.905740174920756 0.099873134580804
0 0.000000050882121 0.198246858793914
for k=1:10, [Q,R]=qr(A);A=R*Q; end
A
A =
16.707493316124758 4.759957185125935 -1.154483851303556
-0.000000000000000 -0.905740179521760 0.099873083698189
0 0.000000000000013 0.198246863397007
diary off