MATH 412 Lecture : MATH 412 2013 Winter Test 2
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Do not turn this page over until instructed: on the top of each exam booklet include the serial number of this paper. If you use more than one booklet, number them. 1. (10 pts) in this problem (but not later ones) a brief justi cation will su ce. (a) a real matrix has the characteristic polynomial (x 1)(x 3)3 and the minimal polynomial (x 1)(x 3)2. What is its jordan form? (b) find the jordan form of the square of that matrix. 2. (25 pts) we will solve the di erential equation u = 2u u. (a) set v = (cid:18) u u (cid:19). Find a matrix a m2(r) such v = av. (b) find an invertible matrix s and a matrix j in jordan form such that a = sj s 1. (c) evaluate exp(at). Let a, b endb(v ) be bounded linear maps from v to itself such that a, b commute.
