Sat Jun 29 13:44:52 PDT 1991
REDUCE 3.4, 15-Jul-91 ...
1: 1:
2: 2:
3: 3: % Tests of eigenfunction/eigenvalue code.
v := mat((1,1,-1,1,0),(1,2,-1,0,1),(-1,2,3,-1,0),
(1,-2,1,2,-1),(2,1,-1,3,0))$
mateigen(v,et);
4 3 2
{{ET - 6*ET + 13*ET + 5*ET - 5,
1,
[ 5*ARBCOMPLEX(1)*(ET - 2) ]
[ ---------------------------- ]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
[ ]
[ 2 ]
[ ARBCOMPLEX(1)*ET*( - ET + 6*ET - 8) ]
[--------------------------------------]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
[ ]
[ ARBCOMPLEX(1)*ET*( - 3*ET + 7) ]
[ -------------------------------- ]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
[ ]
[ 3 2 ]
[ ARBCOMPLEX(1)*(ET - 4*ET + 10) ]
[ ---------------------------------- ]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
[ ]
[ ARBCOMPLEX(1) ]
},
{ET - 2,
1,
[ 0 ]
[ ]
[ 0 ]
[ ]
[ARBCOMPLEX(2)]
[ ]
[ARBCOMPLEX(2)]
[ ]
[ARBCOMPLEX(2)]
}}
eigv := third first ws$
% Now check if the equation for the eigenvectors is fulfilled. Note
% that also the last component is zero due to the eigenvalue equation.
v*eigv-et*eigv;
[ 0 ]
[ ]
[ 4 3 2 ]
[ ARBCOMPLEX(1)*(ET - 6*ET + 13*ET + 5*ET - 5) ]
[ ------------------------------------------------- ]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
[ ]
[ 0 ]
[ ]
[ 4 3 2 ]
[ ARBCOMPLEX(1)*( - ET + 6*ET - 13*ET - 5*ET + 5) ]
[ ---------------------------------------------------- ]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
[ ]
[ 4 3 2 ]
[ 2*ARBCOMPLEX(1)*( - ET + 6*ET - 13*ET - 5*ET + 5) ]
[------------------------------------------------------]
[ 3 2 ]
[ 2*ET - 10*ET + 23*ET + 5 ]
% Example of degenerate eigenvalues.
u := mat((2,-1,1),(0,1,1),(-1,1,1))$
mateigen(u,eta);
{{ETA - 1,2,
[ARBCOMPLEX(3)]
[ ]
[ARBCOMPLEX(3)]
[ ]
[ 0 ]
},
{ETA - 2,1,
[ 0 ]
[ ]
[ARBCOMPLEX(4)]
[ ]
[ARBCOMPLEX(4)]
}}
% Example of a fourfold degenerate eigenvalue with two corresponding
% eigenvectors.
w := mat((1,-1,1,-1),(-3,3,-5,4),(8,-4,3,-4),
(15,-10,11,-11))$
mateigen(w,al);
{{AL + 1,
4,
[ ARBCOMPLEX(5) ]
[ --------------- ]
[ 5 ]
[ ]
[ - 5*ARBCOMPLEX(6) + 7*ARBCOMPLEX(5) ]
[--------------------------------------]
[ 5 ]
[ ]
[ ARBCOMPLEX(5) ]
[ ]
[ ARBCOMPLEX(6) ]
}}
eigw := third first ws;
[ ARBCOMPLEX(5) ]
[ --------------- ]
[ 5 ]
[ ]
[ - 5*ARBCOMPLEX(6) + 7*ARBCOMPLEX(5) ]
EIGW := [--------------------------------------]
[ 5 ]
[ ]
[ ARBCOMPLEX(5) ]
[ ]
[ ARBCOMPLEX(6) ]
w*eigw - al*eigw;
- ARBCOMPLEX(5)*(AL + 1)
MAT((---------------------------),
5
((5*ARBCOMPLEX(6)*AL + 5*ARBCOMPLEX(6) - 7*ARBCOMPLEX(5)*AL
- 7*ARBCOMPLEX(5))/5),
( - ARBCOMPLEX(5)*(AL + 1)),
( - ARBCOMPLEX(6)*(AL + 1)))
% Calculate the eigenvectors and eigenvalue equation.
f := mat((0,ex,ey,ez),(-ex,0,bz,-by),(-ey,-bz,0,bx),
(-ez,by,-bx,0))$
factor om;
mateigen(f,om);
4 2 2 2 2 2 2 2 2 2
{{OM + OM *(EX + EY + EZ + BZ + BY + BX ) + EX *BX
2 2
+ 2*EX*EY*BY*BX + 2*EX*EZ*BZ*BX + EY *BY + 2*EY*EZ*BZ*BY
2 2
+ EZ *BZ ,
1,
2
MAT(((OM *ARBCOMPLEX(7)*EZ + OM*ARBCOMPLEX(7)*( - EX*BY + EY*BX)
3
+ ARBCOMPLEX(7)*BZ*(EX*BX + EY*BY + EZ*BZ))/(OM
2 2 2
+ OM*(EX + EY + BZ ))),
2
(( - OM *ARBCOMPLEX(7)*BY + OM*ARBCOMPLEX(7)*( - EX*EZ + BZ*BX)
3
- (ARBCOMPLEX(7)*EY)*(EX*BX + EY*BY + EZ*BZ))/(OM
2 2 2
+ OM*(EX + EY + BZ ))),
2
((OM *ARBCOMPLEX(7)*BX + OM*ARBCOMPLEX(7)*( - EY*EZ + BZ*BY)
3
+ ARBCOMPLEX(7)*EX*(EX*BX + EY*BY + EZ*BZ))/(OM
2 2 2
+ OM*(EX + EY + BZ ))),
(ARBCOMPLEX(7)))
}}
% Specialize to perpendicular electric and magnetic field.
let ez=0,ex=0,by=0;
% Note that we find two eigenvectors to the double eigenvalue 0
% (as it must be).
mateigen(f,om);
{{OM,
2,
[ ARBCOMPLEX(9)*BX - ARBCOMPLEX(8)*BZ ]
[-------------------------------------]
[ EY ]
[ ]
[ ARBCOMPLEX(8) ]
[ ]
[ 0 ]
[ ]
[ ARBCOMPLEX(9) ]
},
2 2 2 2
{OM + EY + BZ + BX ,
1,
[ - ARBCOMPLEX(10)*EY ]
[ ---------------------- ]
[ BX ]
[ ]
[ - ARBCOMPLEX(10)*BZ ]
[ ---------------------- ]
[ BX ]
[ ]
[ 2 2 2 ]
[ ARBCOMPLEX(10)*(EY + BZ + BX ) ]
[----------------------------------]
[ OM*BX ]
[ ]
[ ARBCOMPLEX(10) ]
}}
% The following has 1 as a double eigenvalue. The corresponding
% eigenvector must involve two arbitrary constants.
j := mat((9/8,1/4,-sqrt(3)/8),
(1/4,3/2,-sqrt(3)/4),
(-sqrt(3)/8,-sqrt(3)/4,11/8));
[ 9 1 - SQRT(3) ]
[ --- --- ------------]
[ 8 4 8 ]
[ ]
[ 1 3 - SQRT(3) ]
J := [ --- --- ------------]
[ 4 2 4 ]
[ ]
[ - SQRT(3) - SQRT(3) 11 ]
[------------ ------------ ---- ]
[ 8 4 8 ]
mateigen(j,x);
{{X - 1,
2,
[SQRT(3)*ARBCOMPLEX(12) - 2*ARBCOMPLEX(11)]
[ ]
[ ARBCOMPLEX(11) ]
[ ]
[ ARBCOMPLEX(12) ]
},
{X - 2,
1,
[ - SQRT(3)*ARBCOMPLEX(13) ]
[ --------------------------- ]
[ 3 ]
[ ]
[ - 2*SQRT(3)*ARBCOMPLEX(13) ]
[-----------------------------]
[ 3 ]
[ ]
[ ARBCOMPLEX(13) ]
}}
% The following is a good consistency check.
sym := mat(
(0, 1/2, 1/(2*sqrt(2)), 0, 0),
(1/2, 0, 1/(2*sqrt(2)), 0, 0),
(1/(2*sqrt(2)), 1/(2*sqrt(2)), 0, 1/2, 1/2),
(0, 0, 1/2, 0, 0),
(0, 0, 1/2, 0, 0))$
ans := mateigen(sym,eta);
ANS := {{ETA,
1,
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ - ARBCOMPLEX(14)]
[ ]
[ ARBCOMPLEX(14) ]
},
{ETA - 1,
1,
[ 2*ARBCOMPLEX(15) ]
[------------------]
[ SQRT(2) ]
[ ]
[ 2*ARBCOMPLEX(15) ]
[------------------]
[ SQRT(2) ]
[ ]
[ 2*ARBCOMPLEX(15) ]
[ ]
[ ARBCOMPLEX(15) ]
[ ]
[ ARBCOMPLEX(15) ]
},
{2*ETA + 1,
1,
[ - ARBCOMPLEX(16)]
[ ]
[ ARBCOMPLEX(16) ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
},
2
{4*ETA + 2*ETA - 1,
1,
[ - ARBCOMPLEX(17) ]
[ ------------------- ]
[ 2*SQRT(2)*ETA ]
[ ]
[ - ARBCOMPLEX(17) ]
[ ------------------- ]
[ 2*SQRT(2)*ETA ]
[ ]
[ ARBCOMPLEX(17)*( - 2*ETA + 1) ]
[-------------------------------]
[ 2*ETA ]
[ ]
[ ARBCOMPLEX(17) ]
[ ]
[ ARBCOMPLEX(17) ]
}}
% Check of correctness for this example.
for each j in ans do
for each k in solve(first j,eta) do
write sub(k,sym*third j - eta*third j);
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
% Tests of nullspace operator.
a1 := mat((1,2,3,4),(5,6,7,8));
[1 2 3 4]
A1 := [ ]
[5 6 7 8]
nullspace a1;
{
[ 1 ]
[ ]
[ 0 ]
[ ]
[ - 3]
[ ]
[ 2 ]
,
[ 0 ]
[ ]
[ 1 ]
[ ]
[ - 2]
[ ]
[ 1 ]
}
b1 := {{1,2,3,4},{5,6,7,8}};
B1 := {{1,2,3,4},{5,6,7,8}}
nullspace b1;
{{1,0,-3,2},{0,1,-2,1}}
% Example taken from a bug report for another CA system.
c1 :=
{{(p1**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
(p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
-((p1**2*p2*(s + z))/(p1**2 + p3**2)), p1*(s + z),
-((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
-((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
(p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)},
{0, 0, 0, 0, 0, 0, 0, 0, 0},
{(p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
(p3**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
-((p1*p2*p3*(s + z))/(p1**2 + p3**2)), p3*(s + z),
-((p2*p3**2*(s + z))/(p1**2 + p3**2)),
-((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
(p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)},
{-((p1**2*p2*(s + z))/(p1**2 + p3**2)), 0,
-((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
-((p1**2*p2**2*(s + 2*z))/((p1**2 + p3**2)*z)), (p1*p2*(s + 2*z))/z,
-((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)),
-((p1*p2*p3*z)/(p1**2 + p3**2)), 0, (p1**2*p2*z)/(p1**2 + p3**2)},
{p1*(s + z), 0, p3*(s + z), (p1*p2*(s + 2*z))/z,
-(((p1**2+p3**2)*(s+ 2*z))/z), (p2*p3*(s + 2*z))/z, p3*z,0, -(p1*z)},
{-((p1*p2*p3*(s + z))/(p1**2 + p3**2)), 0,
-((p2*p3**2*(s + z))/(p1**2 + p3**2)),
-((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)), (p2*p3*(s + 2*z))/z,
-((p2**2*p3**2*(s + 2*z))/((p1**2 + p3**2)*z)),
-((p2*p3**2*z)/(p1**2 + p3**2)), 0, (p1*p2*p3*z)/(p1**2 + p3**2)},
{-((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
-((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
-((p1*p2*p3*z)/(p1**2 + p3**2)),p3*z,-((p2*p3**2*z)/(p1**2 + p3**2)),
-((p3**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))), 0,
(p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))},
{0, 0, 0, 0, 0, 0, 0, 0, 0},
{(p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2), 0,
(p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2),
(p1**2*p2*z)/(p1**2 + p3**2), -(p1*z), (p1*p2*p3*z)/(p1**2 + p3**2),
(p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)), 0,
-((p1**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)))}};
2 2 2 2 2
P1 *( - S*Z - Z + P1 + P2 + P3 )
C1 := {{-------------------------------------,
2 2
P1 + P3
0,
2 2 2 2
P1*P3*( - S*Z - Z + P1 + P2 + P3 )
---------------------------------------,
2 2
P1 + P3
2
- (P1 *P2)*(S + Z)
---------------------,
2 2
P1 + P3
P1*(S + Z),
- (P1*P2*P3)*(S + Z)
-----------------------,
2 2
P1 + P3
2 2 2
- (P1*P3)*(P1 + P2 + P3 )
------------------------------,
2 2
P1 + P3
0,
2 2 2 2
P1 *(P1 + P2 + P3 )
-----------------------},
2 2
P1 + P3
{0,0,0,0,0,0,0,0,0},
2 2 2 2
P1*P3*( - S*Z - Z + P1 + P2 + P3 )
{---------------------------------------,
2 2
P1 + P3
0,
2 2 2 2 2
P3 *( - S*Z - Z + P1 + P2 + P3 )
-------------------------------------,
2 2
P1 + P3
- (P1*P2*P3)*(S + Z)
-----------------------,
2 2
P1 + P3
P3*(S + Z),
2
- (P2*P3 )*(S + Z)
---------------------,
2 2
P1 + P3
2 2 2 2
- P3 *(P1 + P2 + P3 )
--------------------------,
2 2
P1 + P3
0,
2 2 2
P1*P3*(P1 + P2 + P3 )
-------------------------},
2 2
P1 + P3
2
- (P1 *P2)*(S + Z)
{---------------------,
2 2
P1 + P3
0,
- (P1*P2*P3)*(S + Z)
-----------------------,
2 2
P1 + P3
2 2
P1 *P2 *( - S - 2*Z)
----------------------,
2 2
Z*(P1 + P3 )
P1*P2*(S + 2*Z)
-----------------,
Z
2
P1*P2 *P3*( - S - 2*Z)
------------------------,
2 2
Z*(P1 + P3 )
- Z*P1*P2*P3
---------------,
2 2
P1 + P3
0,
2
Z*P1 *P2
-----------},
2 2
P1 + P3
{P1*(S + Z),
0,
P3*(S + Z),
P1*P2*(S + 2*Z)
-----------------,
Z
2 2 2 2
- S*P1 - S*P3 - 2*Z*P1 - 2*Z*P3
--------------------------------------,
Z
P2*P3*(S + 2*Z)
-----------------,
Z
Z*P3,
0,
- Z*P1},
- (P1*P2*P3)*(S + Z)
{-----------------------,
2 2
P1 + P3
0,
2
- (P2*P3 )*(S + Z)
---------------------,
2 2
P1 + P3
2
P1*P2 *P3*( - S - 2*Z)
------------------------,
2 2
Z*(P1 + P3 )
P2*P3*(S + 2*Z)
-----------------,
Z
2 2
P2 *P3 *( - S - 2*Z)
----------------------,
2 2
Z*(P1 + P3 )
2
- Z*P2*P3
-------------,
2 2
P1 + P3
0,
Z*P1*P2*P3
------------},
2 2
P1 + P3
2 2 2
- (P1*P3)*(P1 + P2 + P3 )
{------------------------------,
2 2
P1 + P3
0,
2 2 2 2
- P3 *(P1 + P2 + P3 )
--------------------------,
2 2
P1 + P3
- Z*P1*P2*P3
---------------,
2 2
P1 + P3
Z*P3,
2
- Z*P2*P3
-------------,
2 2
P1 + P3
2 2 2 2
- (Z*P3 )*(P1 + P2 + P3 )
-------------------------------,
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
0,
2 2 2
Z*P1*P3*(P1 + P2 + P3 )
-------------------------------},
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
{0,0,0,0,0,0,0,0,0},
2 2 2 2
P1 *(P1 + P2 + P3 )
{-----------------------,
2 2
P1 + P3
0,
2 2 2
P1*P3*(P1 + P2 + P3 )
-------------------------,
2 2
P1 + P3
2
Z*P1 *P2
-----------,
2 2
P1 + P3
- Z*P1,
Z*P1*P2*P3
------------,
2 2
P1 + P3
2 2 2
Z*P1*P3*(P1 + P2 + P3 )
-------------------------------,
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
0,
2 2 2 2
- (Z*P1 )*(P1 + P2 + P3 )
-------------------------------}}
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
nullspace c1;
- P1
{{1,0,-------,0,0,0,0,0,0},
P3
{0,1,0,0,0,0,0,0,0},
- P1
{0,0,0,1,0,-------,0,0,0},
P3
2 2
P1 + P3
{0,0,0,0,1,-----------,0,0,0},
P2*P3
P3
{0,0,0,0,0,0,1,0,----},
P1
{0,0,0,0,0,0,0,1,0}}
d1 := mat
(((p1**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
(p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
-((p1**2*p2*(s + z))/(p1**2 + p3**2)), p1*(s + z),
-((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
-((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
(p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
(0, 0, 0, 0, 0, 0, 0, 0, 0),
((p1*p3*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2), 0,
(p3**2*(p1**2 + p2**2 + p3**2 - s*z - z**2))/(p1**2 + p3**2),
-((p1*p2*p3*(s + z))/(p1**2 + p3**2)), p3*(s + z),
-((p2*p3**2*(s + z))/(p1**2 + p3**2)),
-((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
(p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
( ((p1**2*p2*(s + z))/(p1**2 + p3**2)), 0,
-((p1*p2*p3*(s + z))/(p1**2 + p3**2)),
-((p1**2*p2**2*(s + 2*z))/((p1**2 + p3**2)*z)), (p1*p2*(s + 2*z))/z,
-((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)),
-((p1*p2*p3*z)/(p1**2 + p3**2)), 0, (p1**2*p2*z)/(p1**2 + p3**2)),
(p1*(s + z), 0, p3*(s + z), (p1*p2*(s + 2*z))/z,
-(((p1**2 + p3**2)*(s + 2*z))/z),(p2*p3*(s + 2*z))/z,p3*z,0,-(p1*z)),
(-((p1*p2*p3*(s + z))/(p1**2 + p3**2)), 0,
-((p2*p3**2*(s + z))/(p1**2 + p3**2)),
-((p1*p2**2*p3*(s + 2*z))/((p1**2 + p3**2)*z)), (p2*p3*(s + 2*z))/z,
-((p2**2*p3**2*(s + 2*z))/((p1**2 + p3**2)*z)),
-((p2*p3**2*z)/(p1**2 + p3**2)), 0, (p1*p2*p3*z)/(p1**2 + p3**2)),
(-((p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)), 0,
-((p3**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2)),
-((p1*p2*p3*z)/(p1**2 + p3**2)),p3*z,-((p2*p3**2*z)/(p1**2 + p3**2)),
-((p3**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))), 0,
(p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z))),
(0, 0, 0, 0, 0, 0, 0, 0, 0),
((p1**2*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2), 0,
(p1*p3*(p1**2 + p2**2 + p3**2))/(p1**2 + p3**2),
(p1**2*p2*z)/(p1**2 + p3**2), -(p1*z), (p1*p2*p3*z)/(p1**2 + p3**2),
(p1*p3*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)), 0,
-((p1**2*(p1**2 + p2**2 + p3**2)*z)/((p1**2 + p3**2)*(s + z)))));
2 2 2 2 2
P1 *( - S*Z - Z + P1 + P2 + P3 )
D1 := MAT((-------------------------------------,0,
2 2
P1 + P3
2 2 2 2
P1*P3*( - S*Z - Z + P1 + P2 + P3 )
---------------------------------------,
2 2
P1 + P3
2
- (P1 *P2)*(S + Z) - (P1*P2*P3)*(S + Z)
---------------------,P1*(S + Z),-----------------------,
2 2 2 2
P1 + P3 P1 + P3
2 2 2 2 2 2 2
- (P1*P3)*(P1 + P2 + P3 ) P1 *(P1 + P2 + P3 )
------------------------------,0,-----------------------),
2 2 2 2
P1 + P3 P1 + P3
(0,0,0,0,0,0,0,0,0),
2 2 2 2
P1*P3*( - S*Z - Z + P1 + P2 + P3 )
(---------------------------------------,0,
2 2
P1 + P3
2 2 2 2 2
P3 *( - S*Z - Z + P1 + P2 + P3 )
-------------------------------------,
2 2
P1 + P3
2
- (P1*P2*P3)*(S + Z) - (P2*P3 )*(S + Z)
-----------------------,P3*(S + Z),---------------------,
2 2 2 2
P1 + P3 P1 + P3
2 2 2 2 2 2 2
- P3 *(P1 + P2 + P3 ) P1*P3*(P1 + P2 + P3 )
--------------------------,0,-------------------------),
2 2 2 2
P1 + P3 P1 + P3
2
P1 *P2*(S + Z) - (P1*P2*P3)*(S + Z)
(----------------,0,-----------------------,
2 2 2 2
P1 + P3 P1 + P3
2 2
P1 *P2 *( - S - 2*Z) P1*P2*(S + 2*Z)
----------------------,-----------------,
2 2 Z
Z*(P1 + P3 )
2 2
P1*P2 *P3*( - S - 2*Z) - Z*P1*P2*P3 Z*P1 *P2
------------------------,---------------,0,-----------),
2 2 2 2 2 2
Z*(P1 + P3 ) P1 + P3 P1 + P3
P1*P2*(S + 2*Z)
(P1*(S + Z),0,P3*(S + Z),-----------------,
Z
2 2 2 2
- S*P1 - S*P3 - 2*Z*P1 - 2*Z*P3 P2*P3*(S + 2*Z)
--------------------------------------,-----------------,
Z Z
Z*P3,0, - Z*P1),
2
- (P1*P2*P3)*(S + Z) - (P2*P3 )*(S + Z)
(-----------------------,0,---------------------,
2 2 2 2
P1 + P3 P1 + P3
2
P1*P2 *P3*( - S - 2*Z) P2*P3*(S + 2*Z)
------------------------,-----------------,
2 2 Z
Z*(P1 + P3 )
2 2 2
P2 *P3 *( - S - 2*Z) - Z*P2*P3 Z*P1*P2*P3
----------------------,-------------,0,------------),
2 2 2 2 2 2
Z*(P1 + P3 ) P1 + P3 P1 + P3
2 2 2
- (P1*P3)*(P1 + P2 + P3 )
(------------------------------,0,
2 2
P1 + P3
2 2 2 2
- P3 *(P1 + P2 + P3 ) - Z*P1*P2*P3
--------------------------,---------------,Z*P3,
2 2 2 2
P1 + P3 P1 + P3
2 2 2 2 2
- Z*P2*P3 - (Z*P3 )*(P1 + P2 + P3 )
-------------,-------------------------------,0,
2 2 2 2 2 2
P1 + P3 S*P1 + S*P3 + Z*P1 + Z*P3
2 2 2
Z*P1*P3*(P1 + P2 + P3 )
-------------------------------),
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
(0,0,0,0,0,0,0,0,0),
2 2 2 2 2 2 2
P1 *(P1 + P2 + P3 ) P1*P3*(P1 + P2 + P3 )
(-----------------------,0,-------------------------,
2 2 2 2
P1 + P3 P1 + P3
2
Z*P1 *P2 Z*P1*P2*P3
-----------, - Z*P1,------------,
2 2 2 2
P1 + P3 P1 + P3
2 2 2
Z*P1*P3*(P1 + P2 + P3 )
-------------------------------,0,
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
2 2 2 2
- (Z*P1 )*(P1 + P2 + P3 )
-------------------------------))
2 2 2 2
S*P1 + S*P3 + Z*P1 + Z*P3
nullspace d1;
{
[0]
[ ]
[1]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
,
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 1 ]
[ ]
[ 0 ]
[ ]
[ - P1 ]
[-------]
[ P3 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
,
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 1 ]
[ ]
[ 2 2 ]
[ P1 + P3 ]
[-----------]
[ P2*P3 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
,
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 0 ]
[ ]
[ 1 ]
[ ]
[ 0 ]
[ ]
[ P3 ]
[----]
[ P1 ]
,
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[1]
[ ]
[0]
}
% The following example, by Kenton Yee, was discussed extensively by
% the sci.math.symbolic newsgroup.
m := mat((e^(-1), e^(-1), e^(-1), e^(-1), e^(-1), e^(-1), e^(-1), 0),
(1, 1, 1, 1, 1, 1, 0, 1),(1, 1, 1, 1, 1, 0, 1, 1),
(1, 1, 1, 1, 0, 1, 1, 1),(1, 1, 1, 0, 1, 1, 1, 1),
(1, 1, 0, 1, 1, 1, 1, 1),(1, 0, 1, 1, 1, 1, 1, 1),
(0, e, e, e, e, e, e, e));
[ 1 1 1 1 1 1 1 ]
[--- --- --- --- --- --- --- 0]
[ E E E E E E E ]
[ ]
[ 1 1 1 1 1 1 0 1]
[ ]
[ 1 1 1 1 1 0 1 1]
[ ]
M := [ 1 1 1 1 0 1 1 1]
[ ]
[ 1 1 1 0 1 1 1 1]
[ ]
[ 1 1 0 1 1 1 1 1]
[ ]
[ 1 0 1 1 1 1 1 1]
[ ]
[ 0 E E E E E E E]
eig := mateigen(m,x);
EIG := {{X - 1,
3,
[ 0 ]
[ ]
[ - ARBCOMPLEX(20)]
[ ]
[ - ARBCOMPLEX(19)]
[ ]
[ - ARBCOMPLEX(18)]
[ ]
[ ARBCOMPLEX(18) ]
[ ]
[ ARBCOMPLEX(19) ]
[ ]
[ ARBCOMPLEX(20) ]
[ ]
[ 0 ]
},
{X + 1,
3,
ARBCOMPLEX(23)
MAT((----------------),
E
(ARBCOMPLEX(22)),
(ARBCOMPLEX(21)),
(( - ARBCOMPLEX(23)*E - ARBCOMPLEX(23)
- 2*ARBCOMPLEX(22)*E - 2*ARBCOMPLEX(21)*E)/(2*E)),
(( - ARBCOMPLEX(23)*E - ARBCOMPLEX(23)
- 2*ARBCOMPLEX(22)*E - 2*ARBCOMPLEX(21)*E)/(2*E)),
(ARBCOMPLEX(21)),
(ARBCOMPLEX(22)),
(ARBCOMPLEX(23)))
},
2 2
{ - E *X + E*X - 6*E*X + 7*E - X,
1,
8 7 7 6
MAT(((6*ARBCOMPLEX(24)*(E *X + 23*E *X - 7*E + 179*E *X
6 5 5 4 4
- 119*E + 565*E *X - 581*E + 768*E *X - 890*E
3 3 2 2
+ 565*E *X - 581*E + 179*E *X - 119*E + 23*E*X
3 8 7 7 6
- 7*E + X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
9 8 8 7
((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E + 310*E *X
7 6 6 5
- 161*E + 1520*E *X - 1246*E + 3577*E *X
5 4 4 3
- 3836*E + 4283*E *X - 4795*E + 2988*E *X
3 2 2
- 3065*E + 978*E *X - 672*E + 132*E*X - 42*E
2 8 7 7 6
+ 6*X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
9 8 8 7
((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E + 310*E *X
7 6 6 5
- 161*E + 1520*E *X - 1246*E + 3577*E *X
5 4 4 3
- 3836*E + 4283*E *X - 4795*E + 2988*E *X
3 2 2
- 3065*E + 978*E *X - 672*E + 132*E*X - 42*E
2 8 7 7 6
+ 6*X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
9 8 8 7
((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E + 310*E *X
7 6 6 5
- 161*E + 1520*E *X - 1246*E + 3577*E *X
5 4 4 3
- 3836*E + 4283*E *X - 4795*E + 2988*E *X
3 2 2
- 3065*E + 978*E *X - 672*E + 132*E*X - 42*E
2 8 7 7 6
+ 6*X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
9 8 8 7
((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E + 310*E *X
7 6 6 5
- 161*E + 1520*E *X - 1246*E + 3577*E *X
5 4 4 3
- 3836*E + 4283*E *X - 4795*E + 2988*E *X
3 2 2
- 3065*E + 978*E *X - 672*E + 132*E*X - 42*E
2 8 7 7 6
+ 6*X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
9 8 8 7
((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E + 310*E *X
7 6 6 5
- 161*E + 1520*E *X - 1246*E + 3577*E *X
5 4 4 3
- 3836*E + 4283*E *X - 4795*E + 2988*E *X
3 2 2
- 3065*E + 978*E *X - 672*E + 132*E*X - 42*E
2 8 7 7 6
+ 6*X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
9 8 8 7
((ARBCOMPLEX(24)*(E *X + 29*E *X - 7*E + 310*E *X
7 6 6 5
- 161*E + 1520*E *X - 1246*E + 3577*E *X
5 4 4 3
- 3836*E + 4283*E *X - 4795*E + 2988*E *X
3 2 2
- 3065*E + 978*E *X - 672*E + 132*E*X - 42*E
2 8 7 7 6
+ 6*X))/(E *(E *X + 30*E *X - 7*E + 333*E *X
6 5 5 4
- 168*E + 1692*E *X - 1365*E + 4023*E *X
4 3 3 2
- 4368*E + 4470*E *X - 5145*E + 2663*E *X
2
- 2520*E + 576*E*X - 251*E + 36*X))),
(ARBCOMPLEX(24)))
}}
% Now check the eigenvectors and calculate the eigenvalues in the
% respective eigenspaces:
factor expt;
for each eispace in eig do
begin scalar eivaleq,eival,eivec;
eival := solve(first eispace,x);
for each soln in eival do
<<eival := rhs soln;
eivec := third eispace;
eivec := sub(soln,eivec);
write "eigenvalue = ", eival;
write "check of eigen equation: ",
m*eivec - eival*eivec>>
end;
eigenvalue = 1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = -1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
4 3 2 2
SQRT(E + 12*E + 10*E + 12*E + 1) + E + 6*E + 1
eigenvalue = ----------------------------------------------------
2*E
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
4 3 2 2
- SQRT(E + 12*E + 10*E + 12*E + 1) + E + 6*E + 1
eigenvalue = -------------------------------------------------------
2*E
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
% For the special choice:
let e = -7 + sqrt 48;
% we get only 7 eigenvectors.
eig := mateigen(m,x);
EIG := {{X + 1,
4,
ARBCOMPLEX(27)
MAT((----------------),
4*SQRT(3) - 7
(ARBCOMPLEX(26)),
(ARBCOMPLEX(25)),
((2*SQRT(3)*( - ARBCOMPLEX(27) - 2*ARBCOMPLEX(26)
- 2*ARBCOMPLEX(25)) + 3*ARBCOMPLEX(27)
+ 7*ARBCOMPLEX(26) + 7*ARBCOMPLEX(25))/(4*SQRT(3) - 7
)),
((2*SQRT(3)*( - ARBCOMPLEX(27) - 2*ARBCOMPLEX(26)
- 2*ARBCOMPLEX(25)) + 3*ARBCOMPLEX(27)
+ 7*ARBCOMPLEX(26) + 7*ARBCOMPLEX(25))/(4*SQRT(3) - 7
)),
(ARBCOMPLEX(25)),
(ARBCOMPLEX(26)),
(ARBCOMPLEX(27)))
},
{X - 1,
3,
[ 0 ]
[ ]
[ - ARBCOMPLEX(30)]
[ ]
[ - ARBCOMPLEX(29)]
[ ]
[ - ARBCOMPLEX(28)]
[ ]
[ ARBCOMPLEX(28) ]
[ ]
[ ARBCOMPLEX(29) ]
[ ]
[ ARBCOMPLEX(30) ]
[ ]
[ 0 ]
},
{X + 7,
1,
[ ARBCOMPLEX(31) ]
[ ----------------- ]
[ 56*SQRT(3) - 97 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
[--------------------------------------------------]
[ 168*SQRT(3) - 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
[--------------------------------------------------]
[ 168*SQRT(3) - 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
[--------------------------------------------------]
[ 168*SQRT(3) - 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
[--------------------------------------------------]
[ 168*SQRT(3) - 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
[--------------------------------------------------]
[ 168*SQRT(3) - 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(31) + 24*ARBCOMPLEX(31) ]
[--------------------------------------------------]
[ 168*SQRT(3) - 291 ]
[ ]
[ ARBCOMPLEX(31) ]
}}
for each eispace in eig do
begin scalar eivaleq,eival,eivec;
eival := solve(first eispace,x);
for each soln in eival do
<<eival := rhs soln;
eivec := third eispace;
eivec := sub(soln,eivec);
write "eigenvalue = ", eival;
write "check of eigen equation: ",
m*eivec - eival*eivec>>
end;
eigenvalue = -1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = 1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = -7
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
% The same behaviour for this choice of e.
clear e;
let e = -7 - sqrt 48;
% we get only 7 eigenvectors.
eig := mateigen(m,x);
EIG := {{X + 1,
4,
- ARBCOMPLEX(34)
MAT((-------------------),
4*SQRT(3) + 7
(ARBCOMPLEX(33)),
(ARBCOMPLEX(32)),
((2*SQRT(3)*( - ARBCOMPLEX(34) - 2*ARBCOMPLEX(33)
- 2*ARBCOMPLEX(32)) - 3*ARBCOMPLEX(34)
- 7*ARBCOMPLEX(33) - 7*ARBCOMPLEX(32))/(4*SQRT(3) + 7
)),
((2*SQRT(3)*( - ARBCOMPLEX(34) - 2*ARBCOMPLEX(33)
- 2*ARBCOMPLEX(32)) - 3*ARBCOMPLEX(34)
- 7*ARBCOMPLEX(33) - 7*ARBCOMPLEX(32))/(4*SQRT(3) + 7
)),
(ARBCOMPLEX(32)),
(ARBCOMPLEX(33)),
(ARBCOMPLEX(34)))
},
{X - 1,
3,
[ 0 ]
[ ]
[ - ARBCOMPLEX(37)]
[ ]
[ - ARBCOMPLEX(36)]
[ ]
[ - ARBCOMPLEX(35)]
[ ]
[ ARBCOMPLEX(35) ]
[ ]
[ ARBCOMPLEX(36) ]
[ ]
[ ARBCOMPLEX(37) ]
[ ]
[ 0 ]
},
{X + 7,
1,
[ - ARBCOMPLEX(38) ]
[ ------------------- ]
[ 56*SQRT(3) + 97 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
[--------------------------------------------------]
[ 168*SQRT(3) + 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
[--------------------------------------------------]
[ 168*SQRT(3) + 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
[--------------------------------------------------]
[ 168*SQRT(3) + 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
[--------------------------------------------------]
[ 168*SQRT(3) + 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
[--------------------------------------------------]
[ 168*SQRT(3) + 291 ]
[ ]
[ - 14*SQRT(3)*ARBCOMPLEX(38) - 24*ARBCOMPLEX(38) ]
[--------------------------------------------------]
[ 168*SQRT(3) + 291 ]
[ ]
[ ARBCOMPLEX(38) ]
}}
for each eispace in eig do
begin scalar eivaleq,eival,eivec;
eival := solve(first eispace,x);
for each soln in eival do
<<eival := rhs soln;
eivec := third eispace;
eivec := sub(soln,eivec);
write "eigenvalue = ", eival;
write "check of eigen equation: ",
m*eivec - eival*eivec>>
end;
eigenvalue = -1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = 1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = -7
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
% For this choice of values
clear e;
let e = 1;
% the eigenvalue 1 becomes 4-fold degenerate. However, we get a complete
% span of 8 eigenvectors.
eig := mateigen(m,x);
EIG := {{X - 1,
4,
[ - ARBCOMPLEX(42)]
[ ]
[ - ARBCOMPLEX(41)]
[ ]
[ - ARBCOMPLEX(40)]
[ ]
[ - ARBCOMPLEX(39)]
[ ]
[ ARBCOMPLEX(39) ]
[ ]
[ ARBCOMPLEX(40) ]
[ ]
[ ARBCOMPLEX(41) ]
[ ]
[ ARBCOMPLEX(42) ]
},
{X + 1,
3,
[ ARBCOMPLEX(45) ]
[ ]
[ ARBCOMPLEX(44) ]
[ ]
[ ARBCOMPLEX(43) ]
[ ]
[ - (ARBCOMPLEX(45) + ARBCOMPLEX(44) + ARBCOMPLEX(43))]
[ ]
[ - (ARBCOMPLEX(45) + ARBCOMPLEX(44) + ARBCOMPLEX(43))]
[ ]
[ ARBCOMPLEX(43) ]
[ ]
[ ARBCOMPLEX(44) ]
[ ]
[ ARBCOMPLEX(45) ]
},
{X - 7,
1,
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
[ ]
[ARBCOMPLEX(46)]
}}
for each eispace in eig do
begin scalar eivaleq,eival,eivec;
eival := solve(first eispace,x);
for each soln in eival do
<<eival := rhs soln;
eivec := third eispace;
eivec := sub(soln,eivec);
write "eigenvalue = ", eival;
write "check of eigen equation: ",
m*eivec - eival*eivec>>
end;
eigenvalue = 1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = -1
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
eigenvalue = 7
check of eigen equation:
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
[ ]
[0]
ma := mat((1,a),(0,b));
[1 A]
MA := [ ]
[0 B]
% case 1:
let a = 0;
mateigen(ma,x);
{{X - 1,1,
[ARBCOMPLEX(47)]
[ ]
[ 0 ]
},
{ - B + X,1,
[ 0 ]
[ ]
[ARBCOMPLEX(48)]
}}
% case 2:
clear a;
let a = 0, b = 1;
mateigen(ma,x);
{{X - 1,2,
[ARBCOMPLEX(49)]
[ ]
[ARBCOMPLEX(50)]
}}
% case 3:
clear a,b;
mateigen(ma,x);
{{ - B + X,
1,
[ ARBCOMPLEX(51)*A ]
[------------------]
[ B - 1 ]
[ ]
[ ARBCOMPLEX(51) ]
},
{X - 1,1,
[ARBCOMPLEX(52)]
[ ]
[ 0 ]
}}
% case 4:
let b = 1;
mateigen(ma,x);
{{X - 1,2,
[ARBCOMPLEX(53)]
[ ]
[ 0 ]
}}
end;
4: 4:
Quitting
Sat Jun 29 13:45:44 PDT 1991