Lab05_slides.sagews

Eigenvalues and Eigenvectors

What is Eigenvalues and Eigenvectors?

eigen-: Characteristic

How to Find it?

#Build the matrix first
matA = matrix(RDF,[[3,1],[6,5]])
matA.right_eigenvectors()

[(1.3542486889354093, [(-0.5192793014075969, 0.8546045911002574)], 1), (6.645751311064591, [(-0.2645215466442612, -0.9643797754831485)], 1)]

Wait, what is RDF?

It means the matrix is in "Real Double Field", which is necessary for sage to calculate eigenvalues and eigenvectors!

eig = matA.right_eigenvectors()
eig[0]
eig[0][1] # can this be a vector?

(1.3542486889354093, [(-0.5192793014075969, 0.8546045911002574)], 1) [(-0.5192793014075969, 0.8546045911002574)]

Some Tricks About Programming

firstEvec = eig[0][1][0] # it was a tuple inside a list!!!!
plot(vector(firstEvec))

Why Eigenvalues and Eigenvectors are Important

matA =matrix(RDF,[[8,1],[6,-5]])
matA
eig = matA.right_eigenvectors()
eig
evec1 = vector(eig[0][1][0])
evec2 = vector(eig[1][1][0])

[ 8.0 1.0] [ 6.0 -5.0] [(8.446221994724901, [(0.9132080913100002, 0.40749353610326905)], 1), (-5.446221994724902, [(-0.07416551254019531, 0.9972459459680194)], 1)]

evec1_2 = matA*evec1
evec2_2 = matA*evec2
fig1 = plot(evec1,color='red')+plot(evec1_2,color='red',linestyle='--',legend_label='evec1_2',legend_color='red')
fig2 = plot(evec2,color='blue')+plot(evec2_2,color='blue',linestyle='--',legend_label='evec2_2',legend_color='blue')
fig1+fig2

evec1_2.norm()
eig[0][0]

8.446221994724901 8.446221994724901

evec2_2.norm()
eig[1][0]

5.446221994724902 -5.446221994724902

Dominate Eigenvalues

matA =matrix(RDF,[[.8,-.1],[-.2,0.8]])
matA.right_eigenvectors()

[(0.9414213562373095, [(0.5773502691896258, -0.8164965809277261)], 1), (0.6585786437626905, [(0.577350269189626, 0.8164965809277259)], 1)]

curState = vector([10,8])
state1 = [curState[0]]
state2 = [curState[1]]
timeSteps = srange(0,100,1)
for i in srange(100):
    nextState = matA*vector([state1[i],state2[i]])
    curState = nextState
    state1.append(nextState[0])
    state2.append(nextState[1])

list_plot(zip(timeSteps,state1))

list_plot(zip(timeSteps,state2))

matA =matrix(RDF,[[.97,-.1],[-.2,0.8]])
eig = matA.right_eigenvectors()
eig

[(1.05, [(0.7808688094430303, -0.6246950475544243)], 1), (0.72, [(0.37139067635410383, 0.9284766908852592)], 1)]

curState = vector([10,8])
result = [[curState[0],curState[1]]]
state1 = [curState[0]]
state2 = [curState[1]]
timeSteps = srange(0,100,1)
for i in srange(100):
    nextState = matA*vector([state1[i],state2[i]])
    resultVec =  matA*vector(result[-1])
    result.append([resultVec[0],resultVec[1]])
    curState = nextState
    state1.append(nextState[0])
    state2.append(nextState[1])
list_plot(zip(timeSteps,state1))
list_plot(zip(timeSteps,state2))
list_plot(zip(timeSteps,result[:][0]))

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags) in namespace, locals File "", line 1, in <module> TypeError: list indices must be integers, not tuple

list_plot(zip(timeSteps,result[:,0]))

Error in lines 1-1 Traceback (most recent call last): File "/cocalc/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 1188, in execute flags=compile_flags) in namespace, locals File "", line 1, in <module> TypeError: list indices must be integers, not tuple

Can we predict more about the result?

evec = vector(eig[0][1][0])
evec

(0.7808688094430303, -0.6246950475544243)

state1[-1]/(abs(state1[-1]+abs(state2[-1])))

0.5555555555555556

evec[0]/(abs(evec[0])+abs(evec[1]))

0.5555555555555555