Exercise 7113 : Let T be a linear operator on a...

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Section 7.1 Exercise 7.1.13 : Let T be a linear operator on a finite-dimensional vector space V such that the charac- teristic polynomial of T splits, and let λ 1 , λ 2 , . . . , λ k be the distinct eigenvalues of T . For each i , let J i be the Jordan canonical form of the restriction of T to K λ i . Prove that J = J 1 J 2 . . . J k is the Jordan canonical form of T . Remark: The statement of the exercise in the book directs you to prove that “ J is the Jordan canonical form of J .” Although that statement actually makes sense (it amounts to saying, “Prove that the matrix J is in Jordan canonical form”), I’m guessing the book intended to state “. . . the Jordan canonical form of T.” The question is also somewhat confusing because it refers to “ the Jordan canonical form,” while the uniqueness of the Jordan canonical form up to ordering of Jordan blocks is not established until the next section of the text. Proof. We first observe that the matrix J is in Jordan canonical form (cf. the remark above). Indeed, by the definition of the direct sum of square matrices on page 320, J is “block diagonal” with diagonal blocks J i . By definition, each J i is in Jordan canonical form; thus, each J i is itself block diagonal with Jordan diagonal blocks. So—by “refining” the block-diagonal decomposition of J —we see that J is a block-diagonal matrix with Jordan diagonal blocks. (Alternatively (and equivalently), note that each J i , being in Jordan canonical form, is a direct sum of Jordan blocks in the sense defined on page 320. Thus J is a direct sum of Jordan blocks and is thus in Jordan canonical form. Strictly speaking, one ought to observe that the direct sum of square matrices is an associative operation, so that ( A B ) C = A B C .) It remains to show that J is a matrix representation of T, i.e. , J = [T] β for some ordered basis β of V . For each 1 i k , let β k be a Jordan canonical basis of K λ i with respect to which J i is the matrix representation of the restriction of T to K λ i . We claim β = β 1 . . . β k is the desired basis of V : We first observe that β is in fact a basis of V . By Theorem 7.8 (which was proven in lecture), we have V = k M i =1 K λ i . Theorem 5.10(d) then shows that β is a basis of V . That J = [T] β then follows from Theorem 5.25. (Writing out a proof of Theorem 5.25 in terms of the “entry-by-entry” definition of page 320 is somewhat instructive but quite tedious. At the least, you should be able to explain to someone in words why that theorem is true.) 1

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