Math 2280-2
\302\2475.4 Multiple Eigenvalue Solutions
Example 5.4.6
with(LinearAlgebra):
A:=Matrix(4,4,[0,0,1,0, 0,0,0,1, -2,2,-3,1, 2,-2,1,-3]);
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
The function of the package LinearAlgebra that computes eigenvectors (note capiltalization in Maple)
Eigenvectors(A);
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
The interpretation of the result we get is that 0 is a simple eigenvalue and -2 is an eigenvalue of algebraic multiplicity 3. Since we could only find 2 eigenvectors associated to lamba=-2, this means the matrix is defective.
The following shows that we only have two eigenvectors associated with lambda=-2
Id:=Matrix(4,4,shape=identity);
NullSpace(A+2*Id);
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
The algebraic multiplicity (multiplicity in the characteristic polynomial) of lambda=-2 is 3. Thus the generalized eigenspace is:
NullSpace((A+2*Id)^3);
PCM8JS0mSSdWZWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0knY29sdW1uRzYiNiMvSSQlaWRHNiIiKHdgLiItJkknVmVjdG9yRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJJ2NvbHVtbkc2IjYjL0kkJWlkRzYiIighW081LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HNiI2Iy9JJCVpZEc2IiIoQy0tIg==
However since the 'defect' or the number of 'missing' eigenvectors is one, we get exactly the same subspace if we look at
NullSpace((A+2*Id)^2);
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
This is because the defect tells you the size of the longest chain you can have. In this case chains no longer than 2 are already captured by NullSpace((A+2*Id)^2).
We take a vector in the generalized eigenspace for lambda=-2
v:=Vector([0,0,1,-1]);
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
We see we obtain a linear combination of the eigenvectors.
u:=(A+2*Id).v;
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Thus (A+2*Id)*u = 0, and we have a chain since (A+2*Id)*v = u.
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