Invariant Subspaces and the Cayley-Hamilton Theorem Michael Freeze MAT 531: Linear Algebra UNC Wilmington Spring 2015 1 / 12 Invariant Subspaces Definition Let T be a linear operator on a vector space V . A subspace W of V is called a T -invariant subspace of V if T (W ) ⊆ W , that is, if T (~v) 2 W for all ~v 2 W . 2 / 12 Invariant Subspaces Definition Let T be a linear operator on a vector space V . A subspace W of V is called a T -invariant subspace of V if T (W ) ⊆ W , that is, if T (~v) 2 W for all ~v 2 W . Example Let T be a linear operator on a vector space V . Then the following subspaces of V are T -invariant: 1. f~0g 2. V 3. R(T ) 4. N(T ) 5. Eλ for any eigenvalue λ of T 3 / 12 Example Let T : R3 ! R3 be defined by T (a; b; c) = (a + b; b + c; 0): What are some T -invariant subspaces of R3? 4 / 12 T -cyclic subspaces Definition Let T be a linear operator on a vector space V and let ~x be a nonzero vector in V . The subspace W = Spanf~x; T (~x); T 2(~x);:::g is called the T -cyclic subspace of V generated by ~x. 5 / 12 Example Let T : R3 ! R3 be defined by T (a; b; c) = (−b + c; a + c; 3c): Determine the T -cyclic subspace generated by e~1 = (1; 0; 0). 6 / 12 Example Let T : P3(R) ! P3(R) be defined by T (f (x)) = f 0(x): Determine the T -cyclic subspace generated by x 2. 7 / 12 Divisibility of the Characteristic Polynomial Theorem Let T be a linear operator on a finite-dimensional vector space V , and let W be a T -invariant subspace of V . Then the characteristic polynomial of TW divides the characteristic polynomial of T . Proof Idea Take an ordered basis γ for W and extend it to an ordered basis β for V . Let B = [TW ]γ. BC Write [T ] = in block form and note that β OD det([T ]β − tI) = det(B − tI) det(D − tI). 8 / 12 Example Let T : R4 ! R4 be defined by T (a; b; c; d) = (a + b + 2c − d; b + d; 2c − d; c + d); and let W = f(t; s; 0; 0) : t; s 2 Rg: Show that W is a T -invariant subspace of R4 and compare the characteristic polynomial of TW to the characteristic polynomial of T . 9 / 12 T -cyclic subspaces and the Characteristic Polynomial Theorem Let T be a linear operator on a finite-dimensional vector space V , and let W denote the T -cyclic subspace of V generated by a nonzero vector ~v 2 V . Let k = dim(W ). Then (a) f~v; T (~v); T 2(~v);:::; T k−1(~v)g is a basis for W . k−1 k (b) If a0~v + a1T (~v) + ··· + ak−1T (~v) + T (~v) = ~0, then the characteristic polynomial of TW is k k−1 k f (t) = (−1) (a0 + a1t + ··· + ak−1t + t ). 10 / 12 The Cayley-Hamilton Theorem Theorem Let T be a linear operator on a finite-dimensional vector space V , and let f (t) be the characteristic polynomial of T . Then f (T ) = T0, the zero transformation. That is, T “satisfies” its characteristic equation. Proof Idea Let ~v 2 V with ~v 6= ~0 and let W be the T -cyclic subspace generated by ~v. Write dim(W ) = k. Use the linear dependence of ~v; T (~v);:::; T k (~v) to obtain the characteristic polynomial of TW and recall that it divides the characteristic polynomial of T . 11 / 12 Example Let T : R2 ! R2 be defined by T (a; b) = (a + 2b; −2a + b): and let β = fe~1; e~2g. Find the characteristic polynomial f (t) of T and verify that f (T ) is the zero transformation. 12 / 12.