Prove Corollaries 1 and 2 of Theorem 5.3

Label the following statements as true or false. (a) Every linear operator on an n-dimensional vector space has n distinct eigenvalues. (b) If a real matrix has one eigenvector, then it has an infinite number of eigenvectors. (c) There exists a square matrix with no eigenvectors. (d) Eigenvalues must be nonzero scalars. (e) Any two eigenvectors are linearly independent. (f) The sum of two eigenvalues of a linear operator T is also an eigenvalue of T. (g) Linear operators on infinite-dimensional vector spaces never have eigenvalues. (h) An n x n matrix A with entries from a field F is similar to a diagonal matrix if and only if there is a basis for Fn consisting of eigenvectors of A. (i) Similar matrices always have the same eigenvalues, (j) Similar matrices always have the same eigenvectors, (k) The sum of two eigenvectors of an operator T is always an eigenvector of T.

For each of the following linear operators T on a vector space V and ordered bases 0, compute [T]/?, and determine whether 0 is a basis consisting of eigenvectors of T. (a) V = R2 , T 10a - 66 17a - 106 , and 0 = (b) V = Pi (R), J(a + 6x) = (6a - 66) + (12a - 116)x, and 3 = {3 + 4x, 2 + 3x} (c) V = R 3a + 26 - 2c' - 4 a - 36 + 2c . and (d) V = P2 (i?),T(a + 6x + cx2 ) = (-4a + 26 - 2c) - (7a + 36 + 7c)x + (7a + 6 + 5c>2 , and 0 = {x - x 2 , - 1 + x 2 , - 1 - x + x 2 }

For each of the following matrices A G MnXn (F), (i) Determine all the eigenvalues of A. (ii) For each eigenvalue A of A, find the set of eigenvectors corresponding to A. If possible, find a basis for Fn consisting of eigenvectors of A. If successful in finding such a basis, determine an invertible matrix Q and a diagonal matrix D such that Q~x AQ = D. = R for F = R (hi; (iv; for F = C for F = R

For each linear operator T on V, find the eigenvalues of T and an ordered basis 0 for V such that [T]/3 is a diagonal matrix. (a) V = R2 and T(a, 6) = (-2a + 36. - 10a + 96) (b) V = R3 and T(a, 6, c) = (7a - 46 + 10c, 4a - 36 + 8c, -2 a + 6 - 2c) (c) V = R3 and T(a, 6, c) = (-4a + 36 - 6c. 6a - 76 + 12c, 6a - 66 + lie) (d) V = P,(7?) and T(u.r + 6) = (-6a + 26)x + (-6a + 6) (e) V = P2(R) and T(/(x)) = x/'(x) + /(2)x + /(3) (f) V = P3(R) and T(/(x)) = f(x) + /(2)x (g) V = P3(R) and T(/(x)) = xf'(x) + f

Prove Theorem 5.4.

Let T be a linear operator on a finite-dimensional vector space V, and let 0 be an ordered basis for V. Prove that A is an eigenvalue of T if and only if A is an eigenvalue of [T]#.

Let T be a linear operator on a finite-dimensional vector space V. We define the determinant of T, denoted det(T), as follows: Choose any ordered basis 0 for V, and define det(T) = det([T]/3). (a) Prove that the preceding definition is independent of the choice of an ordered basis for V. That is, prove that if 0 and 7 are two ordered bases for V, then det([T]/3) = det([T]7). (b) Prove that T is invertible if and only if det(T) ^ 0. (c) Prove that if T is invertible, then det(T_ 1 ) = [det(T)]"1 . (d) Prove that if U is also a linear operator on V, then det(TU) = dct(T)-det(U). (e) Prove that det(T Aly) = det([T]/j XI) for any scalar A and any ordered basis 0 for V.

(a) Prove that a linear operator T on a finite-dimensional vector space is invertible if and only if zero is not an eigenvalue of T. (b) Let T be an invertible linear operator. Prove that a scalar A is an eigenvalue of T if and only if A~! is an eigenvalue of T- 1 . (c) State and prove results analogous to (a) and (b) for matrices.

Prove that the eigenvalues of an upper triangular matrix M are the diagonal entries of M.

Let V be a finite-dimensional vector space, and let A be any scalar. (a) For any ordered basis 0 for V, prove that [Aly]/? = XI. (b) Compute the characteristic polynomial of Aly. (c) Show that Aly is diagonalizable and has only one eigenvalue.

A scalar matrix is a square matrix of the form XI for some scalar A; that is, a scalar matrix is a diagonal matrix in which all the diagonal entries are equal. (a) Prove that if a square matrix A is similar to a scalar matrix XI, then A- XL (b) Show that a diagonalizable matrix having only one eigenvalue is a scalar matrix.

(a) Prove that similar matrices have the same characteristic polynomial. (b) Show that the definition of the characteristic polynomial of a linear operator on a finite-dimensional vector space V is independent of the choice of basis for V.

Let T be a linear operator on a finite-dimensional vector space V over a field F, let 0 be an ordered basis for V, and let A = [T]/3. In reference to Figure 5.1, prove the following. (a) If r G V and 0(v) is an eigenvector of A corresponding to the eigenvalue A, then v is an eigenvector of T corresponding to A. (b) If A is an eigenvalue of A (and hence of T), then a vector y G Fn is an eigenvector of A corresponding to A if and only if 4>~Q (y) is an eigenvector of T corresponding to A

For any square matrix A, prove that A and A1 have the same characteristic polynomial (and hence the same eigenvalues)

(a) Let T be a linear operator on a vector space V, and let x be an eigenvector of T corresponding to the eigenvalue A. For any positive integer m, prove that x is an eigenvector of T m corresponding to the eigenvalue A7 ", (b) State and prove the analogous result for matrices.

(a) Prove that similar matrices have the same trace. Hint: Use Exercise 13 of Section 2.3. (b) How would you define the trace of a linear operator on a finitedimensional vector space? Justify that your definition is welldefined.

Let T be the linear operator on MnXn(i?) defined by T(A) = A1 . (a) Show that 1 are the only eigenvalues of T. (b) Describe the eigenvectors corresponding to each eigenvalue of T. (c) Find an ordered basis 0 for M2x2(Z?) such that [T]/3 is a diagonal matrix. (d) Find an ordered basis 0 for Mnx(R) such that [T]p is a diagonal matrix for n > 2.

Let A,B e M x n (C). (a) Prove that if B is invertible, then there exists a scalar c G C such that A + cB is not invertible. Hint: Examine det(A + cB).

Let A and B be similar nxn matrices. Prove that there exists an ndimensional vector space V, a linear operator T on V, and ordered bases 0 and 7 for V such that A = [T]/3 and B = [T]7. Hint: Use Exercise 14 of Section 2.5.

Let A be an nxn matrix with characteristic polynomial f(t) = (-l)n t n + an-ir"- 1 + + a ^ + a0. Prove that /(0) = an = det(A). Deduce that A is invertible if and only if a0 ^ 0.

Let A and f(t) be as in Exercise 20. (a) Prove that/( 0 = (An-t)(A22-t) (Ann-t) + q(t), where q(t) is a polynomial of degree at most n 2. Hint: Apply mathematical induction to n. (b) Show that tr(A) = (-l) n - 1 a n _ i.

(a) Let T be a linear operator on a vector space V over the field F, and let g(t) be a polynomial with coefficients from F. Prove that if x is an eigenvector of T with corresponding eigenvalue A, then g(T)(x) = g(X)x. That is, x is an eigenvector of o(T) with corresponding eigenvalue g(X). (b) State and prove a comparable result for matrices. (c) Verify (b) for the matrix A in Exercise 3(a) with polynomial g(t) = 2t2 t + 1, eigenvector x = I J, and corresponding eigenvalue A = 4.

Use Exercise 22 to prove that if f(t) is the characteristic polynomial of a diagonalizable linear operator T, then /(T) = To, the zero operator. (In Section 5.4 we prove that this result does not depend on the diagonalizability of T.)

Use Exercise 21(a) to prove Theorem 5.3.

Prove Corollaries 1 and 2 of Theorem 5.3

Determine the number of distinct characteristic polynomials of matrices in M2x2(Z2).