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charpoly-echelon-eigenvectors-factor-matrix

A:matrix([1,1,-1],[2,...

eigenvectors(A);

charpoly(A,x);

Calculate

charpoly-matrix

s:matrix([36,-36,0,-1...

charpoly(s,x);

Calculate

charpoly-coeff-matrix-ratexpand

A:matrix([2,1+t,0],[1...

expr:charpoly(A,x);

expr2:ratexpand(expr);

Calculate

charpoly-determinant-expand-lambda-load-matrix

load ("eigen");

A:matrix([-A1-c,-A3,-...

determinant(A);

Calculate

charpoly-matrix

a:matrix([1,3],[1,-1]);

pol:charpoly(a,t);

Calculate

charpoly-coeff-expand-matrix-solve

m: matrix([-3*s1-2*s2...

cpc: expand(charpoly(...

eq1: coeff(cp, x, 0);

Calculate

charpoly-expand-matrix

A:matrix([2,1+t,1],[1...

charpoly(A,x);

expand(%o2);

Calculate

charpoly-matrix

M:matrix([5,1,1,-1],[...

charpoly(M,x,y,z);

Calculate

charpoly-eigenvalues-invert-matrix

M: matrix([1/2, -1/4]...

I: matrix([1, 0],[0,1]);

eigenvalues(M);

Calculate

charpoly-echelon-eigenvectors-matrix

A:matrix([3,-1,1],[7,...

eigenvectors(A);

charpoly(A,x);

Calculate

charpoly

Run Example
(%i1)m:matrix([1,1,-1],[1,5,1],[1,-1,-2]);
                                [ 1   1   - 1 ]
                                [             ]
(%o1)                           [ 1   5    1  ]
                                [             ]
                                [ 1  - 1  - 2 ]
(%i2) eq1:x+y-z=0;
(%o2)                           - z + y + x = 0
(%i3) eq2:x+5*y+z=0;
(%o3)                           z + 5 y + x = 0
(%i4) eq3:x-y-2*z=0;
(%o4)                          - 2 z - y + x = 0
(%i5) linsolve([eq1,eq2,eq3],[x,y,z]);

solve: dependent equations eliminated: (3)
                             3 %r1        %r1
(%o5)                   [x = -----, y = - ---, z = %r1]
                               2           2
(%i6) n:matrix([3/2],[-(1/2)],[1]);
                                    [  3  ]
                                    [  -  ]
                                    [  2  ]
                                    [     ]
(%o6)                               [   1 ]
                                    [ - - ]
                                    [   2 ]
                                    [     ]
                                    [  1  ]
(%i7) m.n;
                                     [ 0 ]
                                     [   ]
(%o7)                                [ 0 ]
                                     [   ]
                                     [ 0 ]
(%i8) charpoly(m,x);
(%o8)                 ((- x - 2) (5 - x) + 1) (1 - x) + 9
(%i9) 
Run Example
m: matrix ([-2,0,2],[3,-3,0],[8,-6,-2]);
                               [ - 2   0    2  ]
                               [               ]
(%o1)                          [  3   - 3   0  ]
                               [               ]
                               [  8   - 6  - 2 ]
(%i2) Pol: charpoly (m,t);
                                                             2
(%o2)            2 (- 8 (- t - 3) - 18) + (- t - 3) (- t - 2)
(%i3) ratsimp(Pol);
                                     3      2
(%o3)                             - t  - 7 t
(%i4) solve(Pol,t);
(%o4)                          [t = - 7, t = 0]
(%i5) eigenvectors(m);
                                          3    5
(%o5)        [[[- 7, 0], [1, 2]], [[[1, - -, - -]], [[1, 1, 1]]]]
                                          4    2
(%i6) 
Run Example
mat: matrix([F1*gamma,F2*gamma,F3*gamma,0,0],[0,0,0,S*e1,0],[0,0,0,0,S*e2],[F1*(1-gamma),F2*(1-gamma),F3*(1-gamma),0,0],[0,0,0,S*(1-e1),S*(1-e2)]);
(%o1) 
     [    F1 gamma        F2 gamma        F3 gamma         0           0      ]
     [                                                                        ]
     [       0               0               0            e1 S         0      ]
     [                                                                        ]
     [       0               0               0             0          e2 S    ]
     [                                                                        ]
     [ F1 (1 - gamma)  F2 (1 - gamma)  F3 (1 - gamma)      0           0      ]
     [                                                                        ]
     [       0               0               0         (1 - e1) S  (1 - e2) S ]
(%i2) cp: charpoly(mat,%lambda);
(%o2) (%lambda e1 F2 S ((1 - e2) S - %lambda) (1 - gamma)
                            2                      2
 - %lambda ((1 - e1) e2 F3 S  (1 - gamma) + %lambda  ((1 - e2) S - %lambda)))
                                                   2
 (F1 gamma - %lambda) + %lambda (1 - e1) e2 F1 F3 S  (1 - gamma) gamma
 - %lambda e1 F1 F2 S ((1 - e2) S - %lambda) (1 - gamma) gamma
(%i3) sol: substitute([F1=8,F2=12,F3=18,S=0.4,e1=0.1,e2=0.02,gamma=0.0725],cp);
(%o3) (0.58 - %lambda) (0.4452 (0.392 - %lambda) %lambda
                                     2
 - %lambda ((0.392 - %lambda) %lambda  + 0.0480816))
 - 0.258216 (0.392 - %lambda) %lambda + 0.027887328 %lambda
(%i4) allroots(sol);
(%o4) [%lambda = 0.0, %lambda = - 1.0976066942388965E-16, 
%lambda = - 0.39421038450343, %lambda = 0.30113872422923, 
%lambda = 1.065071660274204]
(%i5) sol2: substitute([F1=8,F2=12,F3=18,S=0.4,e1=0.1,e2=0.02,%lambda=1],cp);
(%o5) (0.608 - 0.34368 (1 - gamma)) (8 gamma - 1) + 2.74944 (1 - gamma) gamma
(%i6) solve(sol2);

rat: replaced 2.74944 by 8592/3125 = 2.74944

rat: replaced 0.608 by 76/125 = 0.608

rat: replaced -0.34368 by -1074/3125 = -0.34368
                                          59
(%o6)                           [gamma = ----]
                                         1009
(%i7) float(sol2);
(%o7) (0.608 - 0.34368 (1.0 - 1.0 gamma)) (8.0 gamma - 1.0)
                                              + 2.74944 (1.0 - 1.0 gamma) gamma
(%i8) float(59/1009);
(%o8)                          0.058473736372646
(%i9) 
[allroots,at,charpoly,float,lambda,matrix,solve] [allroots,at,charpoly,lambda,matrix,solve] [allroots,at,charpoly,lambda,matrix] [at,charpoly,lambda,matrix] [batch,charpoly,determinant,eigenvalues,eigenvectors,ident,invert,load,matrix] [charpoly,coeff,determinant,diagmatrix,ev,matrix,ratexpand,solve,transpose] [charpoly,coeff,determinant,diagmatrix,matrix,ratexpand,solve,transpose] [charpoly,coeff,eliminate,expand,matrix,plot2d,sublis] [charpoly,coeff,eliminate,expand,matrix,sublis] [charpoly,coeff,expand,matrix,solve,sublis] [charpoly,coeff,expand,matrix] [charpoly,coeff,matrix,ratexpand,solve] [charpoly,coeff,matrix,ratexpand] [charpoly,coeff,matrix] [charpoly,determinant,eigenvalues,eigenvectors,ident,invert,load,matrix] [charpoly,determinant,eigenvalues,ident,invert,load,matrix] [charpoly,determinant,eigenvalues,invert,load,matrix] [charpoly,determinant,expand,load,matrix] [charpoly,determinant,factor,ident,lu_factor,matrix,minor,nullspace,rank] [charpoly,determinant,invert,matrix] [charpoly,diagmatrix,ev,expand,invert,matrix,transpose] [charpoly,echelon,eigenvalues,eigenvectors,factor,matrix] [charpoly,echelon,eigenvectors,factor,matrix] [charpoly,echelon,eigenvectors,invert,matrix,ratsimp,solve] [charpoly,echelon,eigenvectors,matrix,ratsimp,solve] [charpoly,eigenvalues,eigenvectors,expand,factor,matrix,solve] [charpoly,eigenvalues,eigenvectors,factor,matrix,solve] [charpoly,eigenvalues,eigenvectors,matrix,solve] [charpoly,eigenvalues,eigenvectors,matrix] [charpoly,eigenvalues,expand,gramschmidt,invert,load,matrix,sqrt,transpose] [charpoly,eigenvalues,expand,invert,matrix,solve] [charpoly,eigenvalues,expand,invert,matrix] [charpoly,eigenvalues,expand,load,matrix,sqrt] [charpoly,eigenvalues,expand,matrix] [charpoly,eigenvalues,load,matrix,solve] [charpoly,eigenvalues,matrix] [charpoly,eigenvectors,expand,factor,matrix] [charpoly,eigenvectors,invert,matrix,ratsimp,solve] [charpoly,eigenvectors,matrix,nullspace,rank] [charpoly,eigenvectors,matrix,ratsimp,solve] [charpoly,eigenvectors,matrix] [charpoly,expand,factor,matrix] [charpoly,expand,invert,matrix] [charpoly,expand,matrix] [charpoly,factor] [charpoly,matrix,ratsimp,solve] [charpoly,matrix,ratsimp] [charpoly,matrix,solve] [charpoly,matrix] [charpoly]

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