John Travis
Mississippi College
A = matrix([[0, 4], [-1, 0]])
Values = A.eigenvalues()
for value in Values:
show(value)
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A = matrix([[1, 3], [3, 1]])
Vectors = A.eigenvectors_left()
for vector in Vectors:
show(vector[1])
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D, V = A.eigenmatrix_right()
print "D ="
show(D)
print "V ="
show(V)
print "VD ="
show(V*D)
print "AV ="
show(A*V)
print "Testing AV = VD"
V*D == A*V
D = V = VD = AV = Testing AV = VD True D = V = VD = AV = Testing AV = VD True |
# Eigenvalues with multiplicity > 1 still yields generalized eigenvector
A = matrix(QQ,[[1, -3, 2, -1], [-3, 9, -6, 3], [2, -6, 4, -2], [-1, 3, -2, 1]])
D, V = A.eigenmatrix_right()
print "D ="
show(D)
print "V ="
show(V)
print "VD ="
show(V*D)
print "AV ="
show(A*V)
print "Testing AV = VD"
V*D == A*V
D = V = VD = AV = Testing AV = VD True D = V = VD = AV = Testing AV = VD True |
# Canonical example of trouble if eps is close to machine roundoff.
eps = 10^(-15)
A = matrix(QQ,[[3, -2, -.9, 2*eps],[-2, 4, 1, -1*eps],[-1*eps/4, eps/2, -1, 0],[-.5, -.5, .1, 1]])
show(A)
D, V = A.eigenmatrix_right()
print "D ="
show(D)
print "V ="
show(V)
print "VD ="
show(V*D)
print "AV ="
show(A*V)
D = V = VD = AV = D = V = VD = AV = |
A = random_matrix(RDF,500)
Values = A.eigenvalues()
w = [(i, abs(Values[i])) for i in range(len(Values))] # magnitude of each eigenvalue only
show(points(Values))
show(points(w))
 
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show(Values)
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