### Math 818: Problem set 10, due April 8

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, April 8, 2005. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)

Assume all rings are commutative with 1 not equal to 0 unless specifically stated otherwise. Assume that k[t] is a polynomial ring over a field k.
1. Let f(t) be a polynomial in k[t] of degree at least 1. Show that the vector space dimension of V = k[t]/(f(t)) is deg(f).
2. Let f(t) be a nonzero monic polynomial in k[t].
• (a) Let T: k[t]/(f(t)) -> k[t]/(f(t)) be the linear operator given by multiplication by t. Show that the characteristic polynomial p(t) of T is f(t). (See p. 122 for the definition.) [Hint: find the matrix for T in terms of the basis given by the powers of t.]
• (b) Now let f1(t), ..., fr(t) be nonzero monic polynomials in k[t]. Let V = k[t]/(f1(t)) x ... x k[t]/(fr(t)). Let T: V -> V be the linear operator given by multiplication by t. Show that the characteristic polynomial p(t) of T is the product f1(t)...fr(t). (You may assume the fact that the determinant of a block diagonal matrix is the product of the determinants of the blocks.)
3. Let p(t) be the characteristic polynomial p(t) of an n x n matrix A with entries in a field k. Let V be kn, regarded as a k[t]-module via the action of A.
• (a) If f1(t), ..., fr(t) are the invariant factors for A (i.e., monic, positive degree, each one divides the next), with fr(t) being the factor of largest degree, show that fr(A) = 0.
• (b) If f(t) is any polynomial such that f(A) = 0, show that fr(t) divides f(t). [Note: this is why fr(t) is called the minimal polynomial of A; fr(t) is the nonzero monic polynomial g(t) of minimum possible degree such that g(A) = 0.]
• (c) Conclude from (a) and the previous problem that fr(t) divides p(t) and that p(A) = 0. [This is known as the Cayley-Hamilton theorem.]
4. Let A be an n x n matrix with entries in a field k. Let V be kn, regarded as a k[t]-module via the action of A. Show that tI - A is a presentation matrix for V. [Hint: Note that we can regard A as also giving a k[t]-module map k[t]n -> k[t]n. Let e1, ..., en be the standard basis for k[t]n, and let e'1, ..., e'n be the standard basis for V. Let h: k[t]n -> V be the map h(g1(t)e1 + ... + gn(t)en) = g1(t)e'1 + ... gn(t)e'n. If g1(t)e1 + ... + gn(t)en is in ker(h), it will be helpful to show that g1(A)e1 + ... + gn(A)en = 0. It will also be helpful to note that tiej - Aiej = ti-1(tI - A)ej + ti-2(tI - A)Aej + ... + (tI-A)Ai-1ej.]
5. If A is a 3 x 3 matrix with entries in k and with characteristic polynomial p(t) = (t+1)2(t+2) and minimal polynomial (t+1)(t+2), find the rational canonical form of A. Justify your answer.