Applications of linear algebra, Lay 1.10

980 days ago by pub

a=matrix(RR,3,4,[36,51,13,33,52,34,74,45,0,7,1.1,3]) a 
        
[ 36.0000000000000  51.0000000000000  13.0000000000000 
33.0000000000000]
[ 52.0000000000000  34.0000000000000  74.0000000000000 
45.0000000000000]
[0.000000000000000  7.00000000000000  1.10000000000000 
3.00000000000000]
[ 36.0000000000000  51.0000000000000  13.0000000000000  33.0000000000000]
[ 52.0000000000000  34.0000000000000  74.0000000000000  45.0000000000000]
[0.000000000000000  7.00000000000000  1.10000000000000  3.00000000000000]
a.echelon_form() 
        
[ 1.00000000000000 0.000000000000000 0.000000000000000
0.277223183614433]
[0.000000000000000  1.00000000000000 0.000000000000000
0.391920861637007]
[0.000000000000000 0.000000000000000  1.00000000000000
0.233230880491774]
[ 1.00000000000000 0.000000000000000 0.000000000000000 0.277223183614433]
[0.000000000000000  1.00000000000000 0.000000000000000 0.391920861637007]
[0.000000000000000 0.000000000000000  1.00000000000000 0.233230880491774]
T = matrix(2,[.97,.05,.03,.95]) T 
        
[ 0.970000000000000 0.0500000000000000]
[0.0300000000000000  0.950000000000000]
[ 0.970000000000000 0.0500000000000000]
[0.0300000000000000  0.950000000000000]
population=vector([50,200]) 
       
T*population 
        
(58.5000000000000, 191.500000000000)
(58.5000000000000, 191.500000000000)
T^2 
        
[ 0.942400000000000 0.0960000000000000]
[0.0576000000000000  0.904000000000000]
[ 0.942400000000000 0.0960000000000000]
[0.0576000000000000  0.904000000000000]
T^2*population 
        
(66.3200000000000, 183.680000000000)
(66.3200000000000, 183.680000000000)
T*T*population 
        
(66.3200000000000, 183.680000000000)
(66.3200000000000, 183.680000000000)
T^3*population 
        
(73.5144000000000, 176.485600000000)
(73.5144000000000, 176.485600000000)
T^10*population 
        
(110.096226738739, 139.903773261261)
(110.096226738739, 139.903773261261)
T^100*population 
        
(156.224583738317, 93.7754162616819)
(156.224583738317, 93.7754162616819)
pop200=T^200*population 
       
pop200 
        
(156.249993920127, 93.7500060798709)
(156.249993920127, 93.7500060798709)
T*pop200 
        
(156.249994406517, 93.7500055934811)
(156.249994406517, 93.7500055934811)
list_plot? 
       
list_plot([(T^k*population)[1] for k in [0..200]])+list_plot([(T^k*population)[0] for k in [0..200]]) 
       
line([(k,(T^k*population)[1]) for k in [0..50]]) + \ line([(k,(T^k*population)[0]) for k in [0..50]]) 
       
T.change_ring(RDF).eigenvalues() 
        
[1.0, 0.92]
[1.0, 0.92]
T.change_ring(QQ).eigenvectors_right() 
        
[(1, [
(1, 3/5)
], 1), (23/25, [
(1, -1)
], 1)]
[(1, [
(1, 3/5)
], 1), (23/25, [
(1, -1)
], 1)]