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) ∈ W for all ~v ∈ 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) ∈ W for all ~v ∈ W .
Example Let T be a linear operator on a vector space V . Then the following subspaces of V are T -invariant: 1. {~0} 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 = Span{~x, T (~x), T 2(~x),...}
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 = {(t, s, 0, 0) : t, s ∈ R}.
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 ∈ V . Let k = dim(W ). Then (a) {~v, T (~v), T 2(~v),..., T k−1(~v)} 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 ∈ 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 = {e~1, e~2}.
Find the characteristic polynomial f (t) of T and verify that f (T ) is the zero transformation.
12 / 12