operator. Exercise 1.0.6. (7.1) Recall that R. 2. is an inner product when endowed with the inner. product. h(x1, y1),(x
SWILA PROBLEM SET #11 SECTIONS 7.1-7.3 MONDAY, JUNE 18
Goals for Problem Set: • Learn the various definitions of orthogonal transformations. • Use the properties of the adjoint to determine if operators are self-adjoint. • Deduce properties of orthogonal transformations from the definition. Instructions: The below problems are split up by difficulty level: Easiest, Mediumest, and Hardest. Within each difficulty level, the problems are ordered by how good I think they are to do. In other words, the problems that I think are best to do are listed earlier within each difficulty level. I recommend that you try problems from various sections or from sections that are new to you. For some problems, I write “Good for all”. This means that whichever section the problem is located in, it might be useful for everyone to do it. Warnings: Labeling the difficulty of problems was performed rather imprecisely. Thus, some problems will be mislabeled. The same holds with ordering problems by how good they are to do. I also didn’t really proofread these problems so there may be typos, and I don’t know how to completely solve all of them.
I like the first 5 Easiest problems. I like the first 4 Mediumest problems. I think the first 4 Hardest problems look interesting.
Easiest Problems Exercise 1.0.1. (7.3) (a) Say F = R or C. If T : V → V is an orthogonal transformation, i.e., ||T x|| = ||x|| for all x ∈ V , then every eigenvalue of T has absolute value 1. (b) Give an example of a matrix A ∈ M2 (R) such that each eigenvalue has absolute value 1, but A is not an orthogonal matrix. Hint: Recall that the columns of an orthogonal matrix are orthonormal. Exercise 1.0.2. (Zhang, pg. 62, #3.64) (7.1) For A, B ∈ Mn (C), we define the commutator of A and B to be [A, B] = AB − BA. Show that: (a) A and B are both self-adjoint or both skew-adjoint, then [A, B] is skew-adjoint. (b) If one of A and B is self-adjoint and the other one is skew-adjoint, then [A, B] is selfadjoint. 1
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SECTIONS 7.1-7.3 MONDAY, JUNE 18
Exercise 1.0.3. (Zhang, pg. 117, #5.42)(7.2) Recall that for a linear operator T : V → V , T is an orthogonal transformation if and only if the image of every orthonormal basis of V is also an orthonormal basis of V . The following exercise asks if we can tweak this characterization. Let T be a linear operator on an inner product space V of dimension n. Suppose for all orthogonal bases {v1 , v2 , . . . , vn } of V , {T (v1 ), T (v2 ), . . . , T (vn )} is also an orthogonal basis of V . Is T necessarily an orthogonal transformation? Hint: Note x 7→ λx is a linear operator for all λ 6= 0. Exercise 1.0.4. (Zhang, pg. 77, #4.2)(7.1) Let A and B be n × n self-adjoint matrices. Answer true or false. If the statement is false, make sure to supply a counterexample. (1) A + B is self-adjoint. (2) cA is self-adjoint for every scalar c. (3) ABA is self-adjoint. (4) If AB = 0, then BA = 0. Hint: Exactly one of these is false. Exercise 1.0.5. (Zhang, pg. 117, #5.44)(7.2) Fix a nonzero vector space V . Give an example of a map f : V → V satisfying ||f (u)|| = ||u|| for all u ∈ V that is not a linear operator. Exercise 1.0.6. (7.1) Recall that R2 is an inner product when endowed with the inner product h(x1 , y1 ), (x2 , y2 )i = x1 x2 + y1 y2 . Define the matrix 1 2 A= . 2 1 Define the linear operator T : R2 → R2 by T (x) = Ax. (a) Prove by computation that T is self-adjoint, i.e., show that hT x, yi = hx, T yi
for all x, y ∈ R2 .
(b) Give an example of a basis B such that [T ]B is not self-adjoint. In particular, if we remove the restriction that B is orthonormal, we no longer have the identity [T ]∗B = [T ∗ ]B . Hint: For the second part, just try some other basis that isn’t orthonormal, e.g., B = {(1, 0), (1, 1)}. Exercise 1.0.7. (7.2) Give an example of a matrix A ∈ M2 (R) that is normal, but not orthogonal. More specifically, find A ∈ M2 (R) such that AAt = At A 6= I2 . Exercise 1.0.8. (Zhang, pg. 93, #4.74)(7.3) Let U ∈ Mn (C) be a unitary matrix, i.e., U ∗ U = U U ∗ = I. Show that (a) U t and U¯ are unitary. (b) U V is unitary for every n × n unitary matrix V . (c) The eigenvalues of U are all equal to 1 in absolute value. (d) |U x| = 1 for every unit vector x ∈ Cn . (e) The columns (rows) of U form an orthonormal basis for Cn .
SWILA PROBLEM SET #11
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Do any of the above statements imply that U is unitary? Exercise 1.0.9. (Zhang, pg. 94, #4.82)(7.3) Find all 2 × 2 real orthogonal matrices. Exercise 1.0.10. (Zhang, pg. 95, #4.88)(7.3) Let A ∈ Mn (C). Show that if A is unitary, then so is the matrix 1 A −A √ . 2 A A Mediumest Problems Exercise 1.0.11. (Zhang, pg. 62, #3.64)(7.1) For A, B ∈ Mn (C), we define the commutator of A and B to be [A, B] = AB − BA. Show that if A and B are self-adjoint, then the real part of every eigenvalue of [A, B] is zero. Hint: Show [A, B] is skew-adjoint and ponder upon h[A, B]x, xi. Exercise 1.0.12. (Zhang, pg. 117, #5.41)(7.2) If T : V → V is a map on an inner product space V satisfying hT (x), T (y)i = hx, yi, for all x, y ∈ V, show that T must be a linear transformation. Exercise 1.0.13. (Zhang, pg. 118, #5.48)(7.2) Let T be an orthogonal transformation on an inner product space V . Show that V = W1 ⊕ W2 , where W1 = {x ∈ V : T (x) = x} and W2 = {x − T (x) : x ∈ V }. Exercise 1.0.14. (Zhang, pg. 117, #5.43)(7.2) Let T be a linear operator on an inner product space V of dimension n and let {v1 , v2 , . . . , vn } be an orthogonal basis of V . If ||T (vi )|| = ||vi ||,
i = 1, 2, . . . , n,
is T necessarily an orthogonal transformation? Recall one definition of T being orthogonal is that ||T (u)|| = ||u||, u, v ∈ V. Exercise 1.0.15. (Zhang, pg. 57, #3.34) (7.3) If matrices 0 0 0 1 a 1 A = a 1 b and B = 0 1 0 1 b 1 0 0 2 are similar, what are the values of a and b? Find an orthogonal matrix T ∈ O3 (R), namely, T t T = T T t = I, such that T −1 AT = B. Exercise 1.0.16. (Zhang, pg. 85, #4.39)(7.1) Let A and B be n × n self-adjoint matrices. Give an example for which the eigenvalues of AB are not real. In particular, AB is not necessarily self-adjoint. Exercise 1.0.17. (7.3) (Zhang, pg. 94, #4.76) Show that a square complex matrix U is unitary if and only if the column vectors of U are all of length 1 and | det U | = 1. Do all have unitary matrices have determinant 1? Exercise 1.0.18. (Zhang, pg. 94, #4.77)(7.3) If the eigenvalues of A ∈ Mn (C) are all equal to 1 in absolute value and if ||Ax|| ≤ 1 for all unit vectors x ∈ Cn , show that A is unitary.
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SECTIONS 7.1-7.3 MONDAY, JUNE 18
Exercise 1.0.19. (Zhang, pg. 94, #4.78)(7.3) Show that the n × n Vandermonde matrix U with the (i, j)-entry √1n ω (i−1)(j−1) , where ω n = 1 and ω 6= 1 is symmetric and unitary: 1 1 1 ··· 1 1 ω ω2 · · · ω n−1 4 2n−2 1 1 ω2 ω · · · ω U=√ . .. .. .. .. n .. . . . . . 1 ω n−1 ω 2n−2 · · ·
ω (n−1)
2
Exercise 1.0.20. (Zhang, pg. 95, #4.90)(7.3) Let A be a nonidentity square complex matrix. (a) Can A be self-adjoint and unitary? (b) Can A be upper triangular (but not diagonal) and unitary? Hardest Problems Exercise 1.0.21. (7.2)(Zhang) If S and T are linear operators on an inner product space V such that hS(v), S(v)i = hT (v), T (v)i, v ∈ V, show that there exists an orthogonal operator U such that S = U T. Exercise 1.0.22. (Zhang, pg. 62, #3.64) (7.1) For A, B ∈ Mn (C), we define the commutator of A and B to be [A, B] = AB − BA. Show that if A is a skew-adjoint matrix (A∗ = −A), then A = [B, C] for some self-adjoint matrices B and C. Exercise 1.0.23. (Zhang, pg. 118, #5.45)(7.2) If {v1 , v2 , . . . , vn } and {w1 , w2 , . . . , wn } are two sets of vectors of an inner product space V of dimension n. Note there does necessarily exist a linear transformation that maps each vi to wi for all i (why?). Show that if hvi , vj i = hwi , wj i,
i, j = 1, 2, . . . , n,
then there exists an orthogonal (linear) transformation T such that T (vi ) = wi ,
i = 1, 2, . . . , n.
Exercise 1.0.24. (7.3) (Zhang, pg. 57, #3.36) We say that a matrix A ∈ Mn (C) is unitarily diagonalizable if there exists a unitary matrix U ∈ Mn (C) such that U AU −1 is diagonal. If the eigenvalues of A = (aij ) ∈ Mn (C) are λ1 , λ2 , . . . , λn , show that n X i=1
2
|λi | ≤
n X
|aij |2
i,j=1
and equality holds if and only if A is unitarily diagonalizable. Exercise 1.0.25. (Zhang, pg. 91, #4.62)(7.3) Let A be an n × n complex matrix with rank r. Show that A + A∗ = AA∗ 2 Ir 0 if and only if A = U U ∗ for some unitary matrix U . 0 0
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Exercise 1.0.26. (7.1) (Tao, pg. 49, # 1.4.2) Definition. We say a square matrix A ∈ Mn (C) is positive semidefinite (resp., positive definite) if for all 0 6= x ∈ Cn , hAx, xi ≥ 0 (resp., hAx, xi > 0). Let V be a complex inner product space. If x1 , . . . , xn are a finite collection of vectors in V , show that the Gram matrix (hxi , xj i)1≤i,j≤n is self-adjoint and positive semidefinite, and it is positive definite if and only if the x1 , . . . , xn are linearly independent. Conversely, given a self-adjoint positive semidefinite matrix (aij )1≤i,j≤n with complex entries, show that there exists a complex inner product space V and vectors x1 , . . . , xn such that hxi , xj i = aij for all 1 ≤ i, j ≤ n. Exercise 1.0.27. (Zhang, pg. 82, #4.23)(7.1) Let A = (aij ) be an n × n self-adjoint matrix such that the diagonal entries of A are all equal to 1. If A satisfies n X |aij | ≤ 2, i = 1, 2, . . . , n, j=1
show that (1) A ≥ 0, i.e., x∗ Ax ≥ 0 for all x ∈ Cn . (2) 0 ≤ λ ≤ 2, where λ is any eigenvalue of A. (3) 0 ≤ det A ≤ 1. Exercise 1.0.28. (7.1) (Zhang, pg. 31, # 2.28) Let A, B, C, D ∈ Mn (C). If AB and CD are self-adjoint, show that AD − B ∗ C ∗ = I
⇒
DA − BC = I.
Exercise 1.0.29. (Zhang, pg. 93, #4.74)(hard)(7.3) Let U be an n × n unitary matrix, i.e., U ∗ U = U U ∗ = I. Show that (a) For any k rows of U , 1 ≤ k ≤ n, there exist k columns such that the submatrix formed by the entries on the intersections of these rows and columns is nonsingular. (b) |tr(U A)| ≤ tr(A) for every n × n matrix A ≥ 0. Recall that we say A ≥ 0 if hAx, xi ≥ 0 for all x ∈ Cn . Do any of the above statements imply that U is unitary? Exercise 1.0.30. (Zhang, pg. 95, #4.85)(7.3)(hard) Show that there do not exist real orthogonal matrices A and B satisfying A2 − B 2 = AB. What if “orthogonal” is replaced by “invertible”? References [1] Terence Tao. An Epsilon of Room, I: Real Analysis. American Mathematical Society, Providence, RI, 2010. [2] Fuzhen Zhang. Linear algebra: challenging problems for students. 2nd ed. Johns Hopkins University Press, Baltimore, MD, 2009.
