Suppose T : V V is a linear transformation. Suppose T is diagonalizable (i.e., there is

Decide whether each of the matrices in Exercise 6.1.1 is diagonalizable. Give your reasoning.

Prove or give a counterexample: a. If A is an n n matrix with n distinct (real) eigenvalues, then A is diagonalizable. b. If A is diagonalizable and AB = BA, then B is diagonalizable. c. If there is an invertible matrix P so that A = P 1BP, then A and B have the same eigenvalues. d. If A and B have the same eigenvalues, then there is an invertible matrix P so that A = P 1BP. e. There is no real 2 2 matrix A satisfying A2 = I . f. IfAandB are diagonalizable and have the same eigenvalues (with the same algebraic multiplicities), then there is an invertible matrix P so that A = P 1BP.

Suppose A is a \(2\ \times\ 2\) matrix whose eigenvalues are integers. If det A = 120, explain why A must be diagonalizable.

Consider the differentiation operator D: Pk Pk. Is it diagonalizable?

Let \(f_1(t)=e^t, f_2(t)=t e^t, f_3(t)=t^2 e^t\), and let \(V=\operatorname{Span}\left(f_1, f_2, f_3\right) \subset \mathrm{C}^{\infty}(\mathbb{R})\). Let \(T: V \rightarrow V\) be given by \(T(f)=f^{\prime \prime}-2 f^{\prime}+f\). Decide whether T is diagonalizable.

Is the linear transformation T : Mnn Mnn defined by T (X) = XT diagonalizable? (See Exercise 6.1.12; Exercises 2.5.22 and 3.6.8 may also be relevant.)

Let A = _ 1 1 1 3 . We saw in Example 2 that A has repeated eigenvalue 2 and v1 = _ 1 1 spans E(2). a. Calculate (A 2I)2. b. Solve (A 2I)v2 = v1 for v2. Explain how we know a priori that this equation has a solution. c. Give the matrix for A with respect to the basis {v1, v2}. This is the closest to diagonal one can get and is called the Jordan canonical form of A. Well explore this thoroughly in Section 1 of Chapter 7.

Calculate the (complex) eigenvalues and (complex) eigenvectors of the rotation matrix A = _ cos sin sin cos .

Prove that if is an eigenvalue of A with geometric multiplicity d, then is an eigenvalue of AT with geometric multiplicity d. (Hint: Use Theorem 4.6 of Chapter 3.)

Here you are asked to complete a different proof of Theorem 2.1. a. Show first that is linearly independent. Suppose , and apply to this equation. Use the fact that to deduce and, hence, that . b. Show next that is linearly independent. (Proceed as in part a, applying to the equation.) c. Continue.

Suppose A is an n n matrix with the property that A2 = A. a. Show that if is an eigenvalue of A, then = 0 or = 1. b. Prove that A is diagonalizable. (Hint: See Exercise 3.2.13.)

Suppose is an matrix with the property that . a. Show that if is an eigenvalue of , then or . b. Prove that c. Prove that and deduce that is diagonalizable. (For an application, see Exercise 6 and Exercise 3.6.8.)

Suppose A is diagonalizable, and let pA(t) denote the characteristic polynomial of A. Show that pA(A) = O. This result is a special case of the Cayley-Hamilton Theorem.

This problem gives a generalization of Exercises 11 and 12 for those readers who recall the technique of partial fraction decomposition from their calculus class. Suppose 1, . . . , k R are distinct and f (t) = (t 1)(t 2) (t k). Suppose A is an n n matrix that satisfies the equation f (A) = O. We want to show that A is diagonalizable. a. By considering the partial fractions decomposition 1 f (t) = c1 t 1 + + ck t k , show that there are polynomials f1(t), . . . , fk(t) satisfying 1 = _k j=1 fj (t) and (t j )fj (t) = cjf (t) for j = 1, . . . , k. Conclude thatwecan write I = _k j=1 fj (A), where (A jI)fj (A) = O for j = 1, . . . , k. b. Show that every x Rn can be decomposed as a sum of vectors x = x1 + +xk, where xj E(j ) for j = 1, . . . , k. (Hint: Use the result of part a.) c. Deduce that A is diagonalizable.

Let A be an n n matrix all of whose eigenvalues are real numbers. Prove that there is a basis for Rn with respect to which the matrix for A becomes upper triangular. (Hint: Consider a basis {v1, v_ 2, . . . , v_ n }, where v1 is an eigenvector. Then repeat the argument with a smaller matrix.)

Let A be an orthogonal 3 3 matrix. a. Prove that the characteristic polynomial p(t) has a real root. b. Prove that _Ax_ = _x_ for all x R3 and deduce that only 1 and 1 can be (real) eigenvalues of A. c. Prove that if det A = 1, then 1 must be an eigenvalue of A. d. Prove that if det A = 1 and A _= I , then A : R3 R3 is given by rotation through some angle about some axis. (Hint: First show dim E(1) = 1. Then show that A maps E(1) to itself and use Exercise 2.5.19.) e. (See the remark on p. 218.) Prove that the composition of rotations in R3 is again a rotation.

We say an matrix is nilpotent if for some positive integer . a. Show that 0 is the only eigenvalue of . b. Suppose and . Prove that there is a basis for with respect to which the matrix for becomes (Hint: Choose in , and then define appropriately to end up with this matrix representation. To argue that is linearly independent, you might want to mimic the proof of Theorem 2.1.)

Suppose T : V V is a linear transformation. Suppose T is diagonalizable (i.e., there is a basis for V consisting of eigenvectors of T ). Suppose, moreover, that there is a subspace W V with the property that T (W) W. Prove that there is a basis for W consisting of eigenvectors of T . (Hint: Using Exercise 3.4.17, concoct a basis for V by starting with a basis for W. Consider the matrix for T with respect to this basis. What is its characteristic polynomial?)

In exercise 17-24, an orthogonal set and a vector in Span are given. Use dot products (not system of linear equations) to represent as a linear combination of the vectors in . and

In exercise 17-24, an orthogonal set and a vector in Span are given. Use dot products (not system of linear equations) to represent as a linear combination of the vectors in . and