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the cpoy and paste of matrix will change the rank?

Asked by dcydhb dcydhb on 17 May 2019
Latest activity Commented on by Walter Roberson
on 18 May 2019
i use the following codes to get the matri of A ,
and the rank of A is 10,but when i copy the A and recalculate the rank of a,it runs out the 1,why?
codes are as this
omega=1.795200;
m0=7.234737;
m1=1.816835;
m2=5.349043;
m3=8.732933;
m4=12.024736;
m5=15.265388;
m6=18.476340;
m7=21.668910;
m8=24.849430;
m9=28.021668;
m10=31.187986;
syms a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 alpha0 alpha1 alpha2 alpha3 alpha4 alpha5 alpha6 alpha7 alpha8 alpha9;
h=22;
hgang=22;
agang=2.2497;
dgang= 21.1940;
a=agang/hgang;
d=dgang/hgang;
equation0=alpha0 == (2*a1*sin(d*m1)*besselk(1, a*m1))/(d*m1*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) + (2*a2*sin(d*m2)*besselk(1, a*m2))/(d*m2*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) + (2*a3*sin(d*m3)*besselk(1, a*m3))/(d*m3*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) + (2*a4*sin(d*m4)*besselk(1, a*m4))/(d*m4*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) + (2*a0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/(d*m0*(sinh(2*m0)/(4*m0) + 1/2)^(1/2));
equation1=alpha1 == (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((sin(2*m1)/(4*m1) + 1/2)^(1/2)*(pi^2 - d^2*m1^2)) + (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((sin(2*m2)/(4*m2) + 1/2)^(1/2)*(pi^2 - d^2*m2^2)) + (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((sin(2*m3)/(4*m3) + 1/2)^(1/2)*(pi^2 - d^2*m3^2)) + (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((sin(2*m4)/(4*m4) + 1/2)^(1/2)*(pi^2 - d^2*m4^2)) - (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((sinh(2*m0)/(4*m0) + 1/2)^(1/2)*(pi^2 + d^2*m0^2));
equation2=alpha2 == (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((4*pi^2 + d^2*m0^2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((4*pi^2 - d^2*m2^2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((4*pi^2 - d^2*m3^2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((4*pi^2 - d^2*m4^2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((4*pi^2 - d^2*m1^2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2));
equation3=alpha3 == (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((9*pi^2 - d^2*m1^2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) + (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((9*pi^2 - d^2*m2^2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) + (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((9*pi^2 - d^2*m3^2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) + (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((9*pi^2 - d^2*m4^2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((9*pi^2 + d^2*m0^2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2));
equation4=alpha4 == (2*a0*d*m0*sinh(d*m0)*(besselj(1, a*m0) + bessely(1, a*m0)*i))/((16*pi^2 + d^2*m0^2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (2*a2*d*m2*sin(d*m2)*besselk(1, a*m2))/((16*pi^2 - d^2*m2^2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (2*a3*d*m3*sin(d*m3)*besselk(1, a*m3))/((16*pi^2 - d^2*m3^2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (2*a4*d*m4*sin(d*m4)*besselk(1, a*m4))/((16*pi^2 - d^2*m4^2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (2*a1*d*m1*sin(d*m1)*besselk(1, a*m1))/((16*pi^2 - d^2*m1^2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2));
equation5=a0 == ((sinh(d*m0) - sinh(m0))/(m0*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (alpha0*sinh(d*m0))/(2*a*m0*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) + (pi*alpha1*d*m0*sinh(d*m0)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)*(pi^2 + d^2*m0^2)) - (2*pi*alpha2*d*m0*sinh(d*m0)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 + d^2*m0^2)*besseli(1, a*m2)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) + (3*pi*alpha3*d*m0*sinh(d*m0)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 + d^2*m0^2)*besseli(1, a*m3)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)) - (4*pi*alpha4*d*m0*sinh(d*m0)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 + d^2*m0^2)*besseli(1, a*m4)*(sinh(2*m0)/(4*m0) + 1/2)^(1/2)))/(m0*(besselj(2, a*m0) + bessely(2, a*m0)*i - besselj(1, a*m0)/(a*m0) + (bessely(1, a*m0)*(-i))/(a*m0)));
equation6=a1 == ((sin(d*m1) - sin(m1))/(m1*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) - (alpha0*sin(d*m1))/(2*a*m1*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) - (pi*alpha1*d*m1*sin(d*m1)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)*(pi^2 - d^2*m1^2)) + (2*pi*alpha2*d*m1*sin(d*m1)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m1^2)*besseli(1, a*m2)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) - (3*pi*alpha3*d*m1*sin(d*m1)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m1^2)*besseli(1, a*m3)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)) + (4*pi*alpha4*d*m1*sin(d*m1)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m1^2)*besseli(1, a*m4)*(sin(2*m1)/(4*m1) + 1/2)^(1/2)))/(m1*(besselk(2, a*m1) - besselk(1, a*m1)/(a*m1)));
equation7=a2 == ((sin(d*m2) - sin(m2))/(m2*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (alpha0*sin(d*m2))/(2*a*m2*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (pi*alpha1*d*m2*sin(d*m2)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)*(pi^2 - d^2*m2^2)) + (2*pi*alpha2*d*m2*sin(d*m2)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m2^2)*besseli(1, a*m2)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) - (3*pi*alpha3*d*m2*sin(d*m2)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m2^2)*besseli(1, a*m3)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)) + (4*pi*alpha4*d*m2*sin(d*m2)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m2^2)*besseli(1, a*m4)*(sin(2*m2)/(4*m2) + 1/2)^(1/2)))/(m2*(besselk(2, a*m2) - besselk(1, a*m2)/(a*m2)));
equation8=a3 == ((sin(d*m3) - sin(m3))/(m3*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (alpha0*sin(d*m3))/(2*a*m3*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (pi*alpha1*d*m3*sin(d*m3)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)*(pi^2 - d^2*m3^2)) + (2*pi*alpha2*d*m3*sin(d*m3)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m3^2)*besseli(1, a*m2)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) - (3*pi*alpha3*d*m3*sin(d*m3)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m3^2)*besseli(1, a*m3)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)) + (4*pi*alpha4*d*m3*sin(d*m3)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m3^2)*besseli(1, a*m4)*(sin(2*m3)/(4*m3) + 1/2)^(1/2)))/(m3*(besselk(2, a*m3) - besselk(1, a*m3)/(a*m3)));
equation9=a4 == ((sin(d*m4) - sin(m4))/(m4*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (alpha0*sin(d*m4))/(2*a*m4*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (pi*alpha1*d*m4*sin(d*m4)*(besseli(2, a*m1) + besseli(1, a*m1)/(a*m1)))/(besseli(1, a*m1)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)*(pi^2 - d^2*m4^2)) + (2*pi*alpha2*d*m4*sin(d*m4)*(besseli(2, a*m2) + besseli(1, a*m2)/(a*m2)))/((4*pi^2 - d^2*m4^2)*besseli(1, a*m2)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) - (3*pi*alpha3*d*m4*sin(d*m4)*(besseli(2, a*m3) + besseli(1, a*m3)/(a*m3)))/((9*pi^2 - d^2*m4^2)*besseli(1, a*m3)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)) + (4*pi*alpha4*d*m4*sin(d*m4)*(besseli(2, a*m4) + besseli(1, a*m4)/(a*m4)))/((16*pi^2 - d^2*m4^2)*besseli(1, a*m4)*(sin(2*m4)/(4*m4) + 1/2)^(1/2)))/(m4*(besselk(2, a*m4) - besselk(1, a*m4)/(a*m4)));
[A, b] = equationsToMatrix([equation0, equation1, equation2,equation3,equation4,equation5,equation6, equation7, equation8,equation9], [a0, a1, a2 ,a3, a4 , alpha0 ,alpha1 ,alpha2 ,alpha3 ,alpha4]);
and the rank( A) is 10,
but when i copt the matrix A and calculate the rank of A,it get out the 1,why?
A =...
...
[ - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936, 10045566870117554386053609892956350802656748920209/41815826252695678451841357052067724394233067470848 + (38212912330913546333933271927439582529157556319393*i)/52269782815869598064801696315084655492791334338560, - 10045566870117554386053609892956350802656748920209/62999744934768208151332428452243698833687383638016 - (38212912330913546333933271927439582529157556319393*i)/78749681168460260189165535565304623542109229547520, 1435080981445364912293372841850907257522392702887/14043753724507969458302419437250801878643960184832 + (126953197112669589149279973180862400429094871493*i)/408248654782208414485535448757290752286161633280, - 10045566870117554386053609892956350802656748920209/147735419663058348367521255509111564279802006339584 - (38212912330913546333933271927439582529157556319393*i)/184669274578822935459401569386389455349752507924480, 1, 0, 0, 0, 0]
[ -296349324225323643384292415563/33639933814238323125075836928, -8129216622937086741211209072144761934456574976889/2050170152471208841565167065708702466276170137600, 2709738874312362247070403024048253978152191658963/3656343846794279512703106531818576750862758051840, -8129216622937086741211209072144761934456574976889/25833800520235551318614639869755705905233233182720, 8129216622937086741211209072144761934456574976889/46644477092029352456507624838042939650496381583360, 0, 1, 0, 0, 0]
[ 13522588663836194993980917796725/17631211176203321124428655362048, -3452062385466744459959884618289904701226396322167/2828023964629027589888871736420775202197262565376, -3452062385466744459959884618289904701226396322167/2190653302455404848074155102501142206346409541632, 3497530279094979189422375499787137488577909141/10694139224177094924916120683256758406712655872, -3452062385466744459959884618289904701226396322167/22265362370793135938565296299199059821039682846720, 0, 0, 1, 0, 0]
[ -1559505055508291766081383701859/7325920830152149842646987177984, 40622780053623480499550084142509355499320739442737/164219357544379496541994966008429797107988168704000, -40622780053623480499550084142509355499320739442737/84389506161396227241690508417791717252379181056000, -40622780053623480499550084142509355499320739442737/48660246143575895230895222155200124072170094592000, 40622780053623480499550084142509355499320739442737/234929899370536863903683924455284018633792028672000, 0, 0, 0, 1, 0]
[ 145838977485677560441873737143/1698523909862782370727849885696, -5230845433647037967648954968353982549043744360977/56440789552437185617852909254801587025922228224000, 5230845433647037967648954968353982549043744360977/42998930339625539910443664973947233039634399232000, -15692536300941113902946864905061947647131233082931/61787494954818384222206472262308220955596423168000, -5230845433647037967648954968353982549043744360977/10768506511621049891271613404731891137384546304000, 0, 0, 0, 0, 1]
[ 1, 0, 0, 0, 0, - 19215206817301820201937645933374036236042240/312547901147939322718983515605629276594765441 - (6768960512610041956879930131725477048483840*i)/34727544571993258079887057289514364066085049, 311822173964424859837463003136/4182101353199550343907676068867, -32758260305221388623706849280/507157969485712932497800171043, 1074690585838732255773514006528/17714189487407758346830959806935, -4217261388076414059818656989184/66954719808397516799363753130003]
[ 0, 1, 0, 0, 0, pi*(144059190304636750863338087835057724199218497968354586295107797/2446196724593975793015010125110610682116089828997016170381967360 + (114182725422888477127721339727956710052886909222178814898704217*i)/611549181148493948253752531277652670529022457249254042595491840), (6373987289404659872595495466573082135095775602773*pi)/164792814120371931452773292525785545285435769487360, (1866902816615813251928194064653293767800945463*pi)/15757081295075041034691086819326489962244734976, -(5155874420064302306074166750515442461692706494419*pi)/63455452499346924584569286752621495539451559936000, (13929477699905966940085900270602586319233983248593*pi)/177769867492423376011634069286936746537615622144000]
[ 0, 0, 1, 0, 0, pi*(- 9464989338576010663641040107852943348161756038187711210495130309/334786751097527661443341868614963494529183647077160847702082191360 - (7502043267713731825867055861261399052044045856414964256985623049*i)/83696687774381915360835467153740873632295911769290211925520547840), -(359647875850784208474461056105259465318300911317*pi)/68783230572620007852236149854508416087444800143360, (1491280192527213932355605863890205848162693411339*pi)/13480422184672866986284910016233667451715615457280, (124803259942280896090021788864765346762906143405079*pi)/1091330405129737807036538042638191289801382035456000, -(337177379549110141775193947667076009201637100775613*pi)/4532584749598244496474862383717467767011759095808000]
[ 0, 0, 0, 1, 0, pi*(15646162213207041479098502059587806306703894195417071959648052421/846030272049793392224269349451262918787793849710511133560980635648 + (12401301438212310647408758642027689291628507785318486157217897481*i)/211507568012448348056067337362815729696948462427627783390245158912), (9936948082441062162015806310171888189711714626463*pi)/4384959577658516572786543633942597099272009024536576, -(5282508055207274506629691783028660420071424958195*pi)/225403800818533532313654072351950733940600273371136, (1403447897422781626477159087055068774733031825133*pi)/6932660975349988900289667134781792796272492544000, (3791654838227565187249719560444373602394912477551*pi)/23918033291036642220972615444403915409124753408000]
[ 0, 0, 0, 0, 1, pi*(- 96850311095890057474600622066292417300526063865839919947163193497/7149692360286571710357404675869855841740536003322368113589050408960 - (76764505309227784444080920316982007181254868855370344850327591917*i)/1787423090071642927589351168967463960435134000830592028397262602240), -(2278150693781893501965631214056678334135659812033*pi)/1648966467348438933504394877733030378029235977584640, (726642584284646511025523077002058332515134227547*pi)/59417389819028630748491999550292796349374886051840, -(182435260226832054739117358996020969042983792915701*pi)/3952585887016323880807243202099537361549452640256000, (492880097916447269765789180295077779652228882159047*pi)/1476790770714909314421347029553103020708755144704000]
and also when i use the
fprintf('A=%s;',char(A))
to print A and get that the rank of A is 1,how queer.

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1 Answer

Answer by Walter Roberson
on 18 May 2019
 Accepted Answer

When you copy and paste
- 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936
then those are evaluated as double precision numbers.
14547186153589216317120976241525 approximates as 14547186153589216981797711118336
50328715046552884871112366751744 approximates as itself
11067376412618097943977112897985 approximates as 11067376412618098658996614332416
12582178761638221217778091687936 approximates as itself
With the numerators in the real and imaginary parts both being approximated, it is not uncommon that the rank of the matrix will change.
Remember, just because you can get the Symbolic Toolbox to output a long integer does not mean that regular MATLAB can work with numbers those long without losing precision.
- sym('14547186153589216317120976241525')/sym('50328715046552884871112366751744') - (sym('11067376412618097943977112897985')*i)/sym('12582178761638221217778091687936')

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A =...
...
[ - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936, 10045566870117554386053609892956350802656748920209/41815826252695678451841357052067724394233067470848 + (38212912330913546333933271927439582529157556319393*i)/52269782815869598064801696315084655492791334338560, - 10045566870117554386053609892956350802656748920209/62999744934768208151332428452243698833687383638016 - (38212912330913546333933271927439582529157556319393*i)/78749681168460260189165535565304623542109229547520, 1435080981445364912293372841850907257522392702887/14043753724507969458302419437250801878643960184832 + (126953197112669589149279973180862400429094871493*i)/408248654782208414485535448757290752286161633280, - 10045566870117554386053609892956350802656748920209/147735419663058348367521255509111564279802006339584 - (38212912330913546333933271927439582529157556319393*i)/184669274578822935459401569386389455349752507924480, 1, 0, 0, 0, 0]
[ -296349324225323643384292415563/33639933814238323125075836928, -8129216622937086741211209072144761934456574976889/2050170152471208841565167065708702466276170137600, 2709738874312362247070403024048253978152191658963/3656343846794279512703106531818576750862758051840, -8129216622937086741211209072144761934456574976889/25833800520235551318614639869755705905233233182720, 8129216622937086741211209072144761934456574976889/46644477092029352456507624838042939650496381583360, 0, 1, 0, 0, 0]
[ 13522588663836194993980917796725/17631211176203321124428655362048, -3452062385466744459959884618289904701226396322167/2828023964629027589888871736420775202197262565376, -3452062385466744459959884618289904701226396322167/2190653302455404848074155102501142206346409541632, 3497530279094979189422375499787137488577909141/10694139224177094924916120683256758406712655872, -3452062385466744459959884618289904701226396322167/22265362370793135938565296299199059821039682846720, 0, 0, 1, 0, 0]
[ -1559505055508291766081383701859/7325920830152149842646987177984, 40622780053623480499550084142509355499320739442737/164219357544379496541994966008429797107988168704000, -40622780053623480499550084142509355499320739442737/84389506161396227241690508417791717252379181056000, -40622780053623480499550084142509355499320739442737/48660246143575895230895222155200124072170094592000, 40622780053623480499550084142509355499320739442737/234929899370536863903683924455284018633792028672000, 0, 0, 0, 1, 0]
[ 145838977485677560441873737143/1698523909862782370727849885696, -5230845433647037967648954968353982549043744360977/56440789552437185617852909254801587025922228224000, 5230845433647037967648954968353982549043744360977/42998930339625539910443664973947233039634399232000, -15692536300941113902946864905061947647131233082931/61787494954818384222206472262308220955596423168000, -5230845433647037967648954968353982549043744360977/10768506511621049891271613404731891137384546304000, 0, 0, 0, 0, 1]
[ 1, 0, 0, 0, 0, - 19215206817301820201937645933374036236042240/312547901147939322718983515605629276594765441 - (6768960512610041956879930131725477048483840*i)/34727544571993258079887057289514364066085049, 311822173964424859837463003136/4182101353199550343907676068867, -32758260305221388623706849280/507157969485712932497800171043, 1074690585838732255773514006528/17714189487407758346830959806935, -4217261388076414059818656989184/66954719808397516799363753130003]
[ 0, 1, 0, 0, 0, pi*(144059190304636750863338087835057724199218497968354586295107797/2446196724593975793015010125110610682116089828997016170381967360 + (114182725422888477127721339727956710052886909222178814898704217*i)/611549181148493948253752531277652670529022457249254042595491840), (6373987289404659872595495466573082135095775602773*pi)/164792814120371931452773292525785545285435769487360, (1866902816615813251928194064653293767800945463*pi)/15757081295075041034691086819326489962244734976, -(5155874420064302306074166750515442461692706494419*pi)/63455452499346924584569286752621495539451559936000, (13929477699905966940085900270602586319233983248593*pi)/177769867492423376011634069286936746537615622144000]
[ 0, 0, 1, 0, 0, pi*(- 9464989338576010663641040107852943348161756038187711210495130309/334786751097527661443341868614963494529183647077160847702082191360 - (7502043267713731825867055861261399052044045856414964256985623049*i)/83696687774381915360835467153740873632295911769290211925520547840), -(359647875850784208474461056105259465318300911317*pi)/68783230572620007852236149854508416087444800143360, (1491280192527213932355605863890205848162693411339*pi)/13480422184672866986284910016233667451715615457280, (124803259942280896090021788864765346762906143405079*pi)/1091330405129737807036538042638191289801382035456000, -(337177379549110141775193947667076009201637100775613*pi)/4532584749598244496474862383717467767011759095808000]
[ 0, 0, 0, 1, 0, pi*(15646162213207041479098502059587806306703894195417071959648052421/846030272049793392224269349451262918787793849710511133560980635648 + (12401301438212310647408758642027689291628507785318486157217897481*i)/211507568012448348056067337362815729696948462427627783390245158912), (9936948082441062162015806310171888189711714626463*pi)/4384959577658516572786543633942597099272009024536576, -(5282508055207274506629691783028660420071424958195*pi)/225403800818533532313654072351950733940600273371136, (1403447897422781626477159087055068774733031825133*pi)/6932660975349988900289667134781792796272492544000, (3791654838227565187249719560444373602394912477551*pi)/23918033291036642220972615444403915409124753408000]
[ 0, 0, 0, 0, 1, pi*(- 96850311095890057474600622066292417300526063865839919947163193497/7149692360286571710357404675869855841740536003322368113589050408960 - (76764505309227784444080920316982007181254868855370344850327591917*i)/1787423090071642927589351168967463960435134000830592028397262602240), -(2278150693781893501965631214056678334135659812033*pi)/1648966467348438933504394877733030378029235977584640, (726642584284646511025523077002058332515134227547*pi)/59417389819028630748491999550292796349374886051840, -(182435260226832054739117358996020969042983792915701*pi)/3952585887016323880807243202099537361549452640256000, (492880097916447269765789180295077779652228882159047*pi)/1476790770714909314421347029553103020708755144704000]
means the same as
A = [ - 14547186153589216317120976241525/50328715046552884871112366751744 - (11067376412618097943977112897985*i)/12582178761638221217778091687936, 10045566870117554386053609892956350802656748920209/41815826252695678451841357052067724394233067470848 + (38212912330913546333933271927439582529157556319393*i)/52269782815869598064801696315084655492791334338560, - 10045566870117554386053609892956350802656748920209/62999744934768208151332428452243698833687383638016 - (38212912330913546333933271927439582529157556319393*i)/78749681168460260189165535565304623542109229547520, 1435080981445364912293372841850907257522392702887/14043753724507969458302419437250801878643960184832 + (126953197112669589149279973180862400429094871493*i)/408248654782208414485535448757290752286161633280, - 10045566870117554386053609892956350802656748920209/147735419663058348367521255509111564279802006339584 - (38212912330913546333933271927439582529157556319393*i)/184669274578822935459401569386389455349752507924480, 1, 0, 0, 0, 0];
disp(A)
disp([ -296349324225323643384292415563/33639933814238323125075836928, -8129216622937086741211209072144761934456574976889/2050170152471208841565167065708702466276170137600, 2709738874312362247070403024048253978152191658963/3656343846794279512703106531818576750862758051840, -8129216622937086741211209072144761934456574976889/25833800520235551318614639869755705905233233182720, 8129216622937086741211209072144761934456574976889/46644477092029352456507624838042939650496381583360, 0, 1, 0, 0, 0])
disp([ 13522588663836194993980917796725/17631211176203321124428655362048, -3452062385466744459959884618289904701226396322167/2828023964629027589888871736420775202197262565376, -3452062385466744459959884618289904701226396322167/2190653302455404848074155102501142206346409541632, 3497530279094979189422375499787137488577909141/10694139224177094924916120683256758406712655872, -3452062385466744459959884618289904701226396322167/22265362370793135938565296299199059821039682846720, 0, 0, 1, 0, 0])
disp([ -1559505055508291766081383701859/7325920830152149842646987177984, 40622780053623480499550084142509355499320739442737/164219357544379496541994966008429797107988168704000, -40622780053623480499550084142509355499320739442737/84389506161396227241690508417791717252379181056000, -40622780053623480499550084142509355499320739442737/48660246143575895230895222155200124072170094592000, 40622780053623480499550084142509355499320739442737/234929899370536863903683924455284018633792028672000, 0, 0, 0, 1, 0])
disp([ 145838977485677560441873737143/1698523909862782370727849885696, -5230845433647037967648954968353982549043744360977/56440789552437185617852909254801587025922228224000, 5230845433647037967648954968353982549043744360977/42998930339625539910443664973947233039634399232000, -15692536300941113902946864905061947647131233082931/61787494954818384222206472262308220955596423168000, -5230845433647037967648954968353982549043744360977/10768506511621049891271613404731891137384546304000, 0, 0, 0, 0, 1])
disp([ 1, 0, 0, 0, 0, - 19215206817301820201937645933374036236042240/312547901147939322718983515605629276594765441 - (6768960512610041956879930131725477048483840*i)/34727544571993258079887057289514364066085049, 311822173964424859837463003136/4182101353199550343907676068867, -32758260305221388623706849280/507157969485712932497800171043, 1074690585838732255773514006528/17714189487407758346830959806935, -4217261388076414059818656989184/66954719808397516799363753130003])
disp([ 0, 1, 0, 0, 0, pi*(144059190304636750863338087835057724199218497968354586295107797/2446196724593975793015010125110610682116089828997016170381967360 + (114182725422888477127721339727956710052886909222178814898704217*i)/611549181148493948253752531277652670529022457249254042595491840), (6373987289404659872595495466573082135095775602773*pi)/164792814120371931452773292525785545285435769487360, (1866902816615813251928194064653293767800945463*pi)/15757081295075041034691086819326489962244734976, -(5155874420064302306074166750515442461692706494419*pi)/63455452499346924584569286752621495539451559936000, (13929477699905966940085900270602586319233983248593*pi)/177769867492423376011634069286936746537615622144000])
disp([ 0, 0, 1, 0, 0, pi*(- 9464989338576010663641040107852943348161756038187711210495130309/334786751097527661443341868614963494529183647077160847702082191360 - (7502043267713731825867055861261399052044045856414964256985623049*i)/83696687774381915360835467153740873632295911769290211925520547840), -(359647875850784208474461056105259465318300911317*pi)/68783230572620007852236149854508416087444800143360, (1491280192527213932355605863890205848162693411339*pi)/13480422184672866986284910016233667451715615457280, (124803259942280896090021788864765346762906143405079*pi)/1091330405129737807036538042638191289801382035456000, -(337177379549110141775193947667076009201637100775613*pi)/4532584749598244496474862383717467767011759095808000])
disp([ 0, 0, 0, 1, 0, pi*(15646162213207041479098502059587806306703894195417071959648052421/846030272049793392224269349451262918787793849710511133560980635648 + (12401301438212310647408758642027689291628507785318486157217897481*i)/211507568012448348056067337362815729696948462427627783390245158912), (9936948082441062162015806310171888189711714626463*pi)/4384959577658516572786543633942597099272009024536576, -(5282508055207274506629691783028660420071424958195*pi)/225403800818533532313654072351950733940600273371136, (1403447897422781626477159087055068774733031825133*pi)/6932660975349988900289667134781792796272492544000, (3791654838227565187249719560444373602394912477551*pi)/23918033291036642220972615444403915409124753408000])
disp([ 0, 0, 0, 0, 1, pi*(- 96850311095890057474600622066292417300526063865839919947163193497/7149692360286571710357404675869855841740536003322368113589050408960 - (76764505309227784444080920316982007181254868855370344850327591917*i)/1787423090071642927589351168967463960435134000830592028397262602240), -(2278150693781893501965631214056678334135659812033*pi)/1648966467348438933504394877733030378029235977584640, (726642584284646511025523077002058332515134227547*pi)/59417389819028630748491999550292796349374886051840, -(182435260226832054739117358996020969042983792915701*pi)/3952585887016323880807243202099537361549452640256000, (492880097916447269765789180295077779652228882159047*pi)/1476790770714909314421347029553103020708755144704000])
That is, having a bunch of [] on consecutive rows does not mean that the rows take part in the previous assignment. Not unless there are ;... on the ends of the lines.
so you mean,
A =...
...
[1 2 3]
[4 5 6]
[7 8 10];
rank(A )=1,
and when
A1 =...
...
[1 2 3
4 5 6]
7 8 10];
now rank(A1)=3,
and also we can use
A2 =...
...
[[1 2 3]
[4 5 6]
[7 8 10]];
and now rank(A2) =3,
so it is just not the approximatation but the assignment of the matrix?
A =...
...
[1 2 3]
The assigment ends after that line.
[4 5 6]
There is no assignment happening, and there is no semi-colon, so a vector is built and the result is displayed.
[7 8 10];
There is no assignment happening, and there is a semi-colon on the end of the line, so a vector is built and the result is discarded.
Assignment continues automatically on to a new line under any of three different circumstances:
  1. The active part of the line ends in ... When that happens, the rest of that line is discarded, and the result is as if the next line was there in place of the ... with the line break being ignored . [123 ... line-break 456] is the same as [123 456] not the same as [123; 456]
  2. there is an active [ that has not been balanced by a ] . When that happens, unless the line ended with ... then the line is treated as-if it ended with ;... so [123 line-break 456] is treated the same as [123 ;... line-break 456] which is the same as [123; 456]
  3. there is an active { that has not been balanced by a } . When that happens, unless the line ended with ... then the line is treated as-if it ended with ;... so {123 line-break 456} is treated the same as {123 ;... line-break 456} which is the same as {123; 456}

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