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0.1 Rational Canonical Forms Draft. Do not cite or quote. We have already seen that it is useful and simpler to study linear systems using matrices. But matrices are themselves cumbersome, as they are stuffed with many entries, and it turns out that it's best to focus on the fact that we can view an m × n matrix A as a function n m n T : R ! R ; namely, T (x) = Ax for every column vector x 2 R . The function T , called a linear transformation, has two important properties: T (x + y) = T (x) + T (y), for all x; y; for every number α and vector x, we have T (αx) = αT (x). As we shall see in the course, the matrix A is a way of studying T by choosing a coordinate n system in R ; choosing another coordinate system gives another matrix describing T . Just as in calculus when studying conic sections, choosing a good coordinate system can greatly simplify things: we can put a conic section defined by Ax2 + Bxy + Cy2 + Dx + Ey + F into normal form, and we can easily read off properties of the conic (is it a parabola, hyperbola, or ellipse; what are its foci, etc) from the normal form. The important object of study for us is a linear transformation T , and any choice of coordinate system produces a matrix describing it. Is there a \best" matrix describing T analogous to a normal form for a conic? The answer is yes, and normal forms here are called canonical forms. Now coordinates arise from bases of a vector space, and the following discussion is often called change of basis. 0.1 Rational Canonical Forms We begin by stating some facts which will eventually be treated in the course proper. If you have any questions, please feel free to ask me. Unless we say otherwise, our vector spaces involve real numbers as scalars (although, in several instances, we may want complex numbers), linear transformations T are functions on a vector space V (that is, T maps V to itself), and matrices are square. If T : V ! V is a linear transformation on a vector space V and x = x1; : : : ; xn is a basis of V , then T determines the matrix A = x[T ]x whose ith column consists of the coordinate list of T (xi) with respect to x. If Y is another basis of V , then the matrix B = Y [T ]Y may be different from A, but Theorem 4.3.1 in the book says that A and B are similar; that is, there exists a nonsingular matrix P with B = P AP −1. Theorem 4:3:1: Let T : V ! V be a linear transformation on a vector space V . If x and Y are bases of V , then there is a nonsingular Abstract Algebra Arising from Fermat's Last Theorem Draft. c 2011 iii Draft. Do not cite or quote. matrix P (called a transition matrix), namely, P = Y [1V ]x, so that −1 Y [T ]Y = P x[T ]x P : Conversely, if B = P AP −1, where B; A, and P are n × n matrices n and P is nonsingular, then there is a linear transformation T : R ! n n R and bases x and Y of k such that B = Y [T ]Y and A = x[T ]x. We now consider how to determine whether two given matrices are similar. Definition If A is an r × r matrix and B is an s × s matrix, then their direct sum A ⊕ B is the (r + s) × (r + s) matrix " # A 0 A ⊕ B = : 0 B Definition If g(x) = x + c0, then its companion matrix C(g) is the 1 × 1 s s−1 matrix [−c0]; if s ≥ 2 and g(x) = x + cs−1x + ··· + c1x + c0, then its companion matrix C(g) is the s × s matrix 2 3 0 0 0 ··· 0 −c0 61 0 0 ··· 0 −c 7 6 1 7 6 7 60 1 0 ··· 0 −c2 7 C(g) = 6 7 : 60 0 1 ··· 0 −c3 7 6 7 6. 7 4. 5 0 0 0 ··· 1 −cs−1 Obviously, we can recapture the polynomial g(x) from the last column of the companion matrix C(g). We call a polynomial f(x) monic if the highest power of x in f has coefficient 1. Theorem 0.1 Every n×n matrix A is similar to a direct sum of companion matrices C(g1) ⊕ · · · ⊕ C(gt) in which the gi(x) are monic polynomials and g1(x) j g2(x) j · · · j gt(x): Definition iv Abstract Algebra Arising from Fermat's Last Theorem Draft. c 2011 Draft. Do not cite or quote. 0.1 Rational Canonical Forms A rational canonical form is a matrix R that is a direct sum of companion matrices, R = C(g1) ⊕ · · · ⊕ C(gt); where the gi(x) are monic polynomials with g1(x) j g2(x) j · · · j gt(x). If a matrix A is similar to a rational canonical form C(g1) ⊕ · · · ⊕ C(gt), where g1(x) j g2(x) j · · · j gt(x), then we say that the invariant factors of A are g1(x); g2(x); : : : ; gt(x). Theorem 0.1 says that every n × n matrix is similar to a rational canonical form, and so it has invariant factors. Can a matrix A have more than one list of invariant factors? Theorem 0.2 1. Two n × n matrices A and B are similar if and only if they have the same invariant factors. 2. An n × n matrix A is similar to exactly one rational canonical form. Corollary 0.3 Let A and B be n × n matrices with real entries. If A and B are similar over C, then they are similar over R (i.e., if there is a nonsingular matrix P having complex entries with B = P AP −1, then there is a nonsingular matrix Q having real entries with B = QAQ−1). Definition If T : V ! V is a linear transformation, then an invariant sub- space is, a subspace W of V with T (W ) ⊆ W . Does a linear transformation T on a finite-dimensional vector space V leave any one-dimensional subspaces of V invariant; that is, is there a nonzero vector v 2 V with T (v) = αv for some α? If 2 2 o T : R ! R is rotation by 90 , then its matrix with respect to the 0 −1 standard basis is 1 0 . Now " # " #" # " # x 0 −1 x −y T : 7! = : y 1 0 y x x If v = [ y ] is a nonzero vector and T (v) = αv for some α 2 R, then αx = −y and αy = x; it follows that (α2 +1)x = x and (α2 +1)y = y. 2 Since v 6= 0, α + 1 = 0 and α2 = R. Thus, T has no one-dimensional Abstract Algebra Arising from Fermat's Last Theorem Draft. c 2011 v Draft. Do not cite or quote. 0 −1 invariant subspaces. Note that 1 0 is the companion matrix of x2 + 1. Definition Let V be a vector space and let T : V ! V be a linear transformation. If T v = αv, where α 2 C and v 2 V is nonzero, then α is called an eigenvalue of T and v is called an eigenvector of T for α. Let A be an n × n matrix. If Av = αv, where α 2 k and v 2 kn is a nonzero column, then α is called an eigenvalue of A and v is called an eigenvector of A for α. Rotation by 90o has no (real) eigenvalues. At the other extreme, can a linear transformation have infinitely many eigenvalues? n n Theorem 4.2.1. If T : R ! R is a linear transformation, then there exists a unique n × n matrix A such that T (v) = Av for all n v 2 R . To say that Av = αv for v nonzero is to say that v is a nontrivial solution of the homogeneous system (A−αI)v = 0; that is, A−αI is a singular matrix. But a matrix is singular if and only if its determinant is 0. Definition The characteristic polynomial of an n × n matrix A is pA(x) = det(xI − A) 2 R[x]: Thus, the eigenvalues of an n × n matrix A are the roots of pA(x), a polynomial of degree n, and so A has at most n real eigenvalues. Note that some eigenvalues of A may be complex numbers. Definition The trace of an n × n matrix A = [aij] is n X tr(A) = aii: i=1 Proposition 0.4 If A = [aij] is an n × n matrix having eigenvalues (with multiplicity) α1; : : : ; αn, then X Y tr(A) = − αi and det(A) = αi: i i vi Abstract Algebra Arising from Fermat's Last Theorem Draft. c 2011 Draft. Do not cite or quote. 0.1 Rational Canonical Forms Proof. For any polynomial f(x) with real coefficients, if f(x) = n n−1 x + cn−1x + ··· + c1x + c0 = (x − α1) ··· (x − αn), then cn−1 = P n Q Qn − i αi and c0 = (−1) i αi. In particular, pA(x) = i=1(x − αi), P so that cn−1 = − i αi = −tr(A). Now the constant term of any polynomial f(x) is just f(0); setting x = 0 in pA(x) = det(xI − A) n Q gives pA(0) = det(−A) = (−1) det(A). Hence, det(A) = i αi. Here are some elementary facts about eigenvalues. Corollary 0.5 Let A be an n × n matrix.
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