Simple Eigenvalues - Queen's Universitymast.queensu.ca/~math211/m211oh/m211oh94.pdf · Simple...
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Simple Eigenvalues Definition: An eigenvalue λ of A is called simple if its algebraic multiplicity m A (λ)=1. Remark. Clearly, each simple eigenvalue is regular. Theorem 10: If A is power convergent and 1 is a sim- ple eigenvalue of A, then lim n→∞ A n = E 10 = 1 ~u t ~v |{z} scalar ~u~v t |{z} matrix , where: ~u ∈ E A (1) is any non-zero 1-eigenvector of A, ~v ∈ E A t (1) is any non-zero 1-eigenvector of A t . Remark. Since det(C t ) = det(C ) for any matrix C , it follows that ch A t (t) = ch A (t). Thus: A and A t have the same eigenvalues with the same algebraic multiplicities. (In fact, A and A t are similar, so they even have the same Jordan canonical form.)
Transcript of Simple Eigenvalues - Queen's Universitymast.queensu.ca/~math211/m211oh/m211oh94.pdf · Simple...
Simple Eigenvalues
Definition: An eigenvalue λ of A is called simple ifits algebraic multiplicity mA(λ) = 1.
Remark. Clearly, each simple eigenvalue is regular.
Theorem 10: If A is power convergent and 1 is a sim-ple eigenvalue of A, then
limn→∞
An = E10 =1
~ut~v︸︷︷︸scalar
~u~vt︸︷︷︸matrix
,
where:
~u ∈ EA(1) is any non-zero 1-eigenvector of A,
~v ∈ EAt(1) is any non-zero 1-eigenvector of At.
Remark. Since det(Ct) = det(C) for any matrix C,it follows that
chAt(t) = chA(t).
Thus: A and At have the same eigenvalues with thesame algebraic multiplicities. (In fact, A and At aresimilar, so they even have the same Jordan canonicalform.)
