Math 554 Qualifying ExaminationAugust, 1997In answering any part of a question you may assume the preceding parts.1.LetAbe an 8×8 complex matrix satisfying the following conditions (withIan 8×8 identity matrix):(i) Rank(A+I) = 6, rank(A+I)2= 5, and rank(A+I)k= 4 for allk≥3.(ii) Rank(A-2I) = 7 and rank(A-2I)k= 6 for allk≥2.(iii) Rank(A-3I)k= 6 for allk≥1.Find the Jordan form ofA.2.(a) Prove that any linear transformationTof a finite-dimensionalC-vectorspaceVcan be written in the formT=S+NwhereSis diagonalizable,Nisnilpotent, andSN=NS.(Hint: Look at the Jordan form.)(b) Use (without proof) the equivalence of “Sdiagonalizable” and “the minimalpolynomial ofShas no multiple factors” to show that ifSis diagonalizable and ifWis anS-invariant subspace ofVthen the restriction ofStoWis diagonalizable.(c) LetT=S+Nbe as in (a). LetIbe the identity transformation ofV. Foranyλ∈C, and anym >0, letWλ,mbe the kernel of (T-λI)m. Show that therestriction of (S-λI
This is the end of the preview.
access the rest of the document.