Let F denote a field. JCF=Jordan canonical form, RCF=rational canonical form.
- (50 points) Suppose a real matrix A has the characteristic polynoial
pA(x) = (x
3 − 8)3
(x
2 + 1)2
.
(a) List all possible combinations of Jordan blocks corresponding to each (real and
complex) eigenvalue of A.
(b) Select 4 possible JCFs of A and for each of them, list the corresponding real JCF,
the minimal polynomial, and the RCFs in elementary divisor and in invariant factor
versions. - (50 points) Suppose a real matrix A ∈ M12(R) has the minimal polynomial
mA(x) = (x
2 − 2)2
(x
2 + x + 1)2
.
What are the possible (complex) JCFs of A?
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