209.sagews

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]

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]

T = matrix(2,[.97,.05,.03,.95])
T

[ 0.970000000000000 0.0500000000000000] [0.0300000000000000 0.950000000000000]

population=vector([50,200])

T*population

(58.5000000000000, 191.500000000000)

T^2

[ 0.942400000000000 0.0960000000000000] [0.0576000000000000 0.904000000000000]

T^2*population

(66.3200000000000, 183.680000000000)

T*T*population

(66.3200000000000, 183.680000000000)

T^3*population

(73.5144000000000, 176.485600000000)

T^10*population

(110.096226738739, 139.903773261261)

T^100*population

(156.224583738317, 93.7754162616819)

pop200=T^200*population

pop200

(156.249993920127, 93.7500060798709)

T*pop200

(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]

T.change_ring(QQ).eigenvectors_right()

[(1, [ (1, 3/5) ], 1), (23/25, [ (1, -1) ], 1)]