Due: Wednesday, December 3, at 4 PM Math 512 / Problem Set 11 (three pages) • Study/read: Inner product spaces & Operators on inner product spaces - Ch. 5 & 6 from LADW by Treil. - Ch. 8 & 9 from Linear Algebra by Hoffman & Kunze. (!) Make sure that you understand perfectly the definitions, examples, theorems, etc. R 1 Legendre Polynomials Recall that (f, g) = −1 f(x)g(x)dx is an inner product on C(I, R), where I = [−1, 1]. The system of polynomials (L0(t), L1(t), ... , Ln−1(t)) resulting from the n−1 Gram-Schmidt orthonormalization of the standard basis E := (1, t, ... , t ) of Pn(R) are called the Legendre polynomials. Google the term “Legendre polynomials” and learn about their applications. 1) Answer / do the following: a) Compute L0, L1, L2, L3. 3 b) Write t − t + 1 as a linear combination of L0, L1, L2, L3. c) Show that in general, one has deg(Lk) = k. k d) What is the coefficient of t in Lk(t)? 2) Which of the following pairs of matrices are unitarily equivalent: 0 1 0 2 0 0 0 1 0 1 0 0 1 0! 0 1! a) & b) −1 0 0 & 0 −1 0 c) −1 0 0 & 0 −i 0 0 1 1 0 0 0 1 0 0 0 0 0 1 0 0 i [Hint: Unitarily equivalent matrices have the same characteristic polynomial (WHY), thus the same eigenvalues including multiplicities, the same trace, determinant, etc.] 3) For each of the following matrices over F = R decide whether they are unitarily diago- nalizable, and if so, find the unitary matrix P over R such that P AP −1 is diagonal. 0 1 2 3 ! ! cos x 0 sin x 1 1 1 2 1 2 3 4 a) A = b) A = c) A = 0 −2 0 d) A = 1 1 2 1 2 3 4 5 sin x 0 − cos x 3 4 5 6 4) In each of the matrices below give all the values of the entries such that the matrix are: a) unitary; b) self adjoint; c) normal; d) orthogonal a 1+i a 1 √1 √1 a 11 2 13 2 3 3 13 a a 1−i a 1 12 22 2 24 A = a21 a22 A = 3 a31 a32 a33 a34 a31 a32 a33 1 1 1 1 2 2 2 2 1 • In the next problems, if not otherwise explicitly stated, V is a finite dimensional F -vector space with inner product, where F = R or F = C. 5) True of false (justify if not obvious): a) Every orthogonal system A = (v1, ... , vr) of vector from V is linearly independent. b) Every orthonormal system A = (v1, ... , vr) of vectors from V is linearly independent. c) If B = (w1, ... , wn) is a basis of V, there exists an orthonormal basis A = (v1, ... , vn) such that hhhhw1, ... , wriiiiF = hhhhv1, ... , vriiiiF for each 1 ≤ r ≤ n. d) For a system of vectors A = (v1, ... , vn) the following are equivalent: i) A is an orthonormal basis of V . ii) For every v ∈ V one has: v = (v, v1)v1 + ··· + (v, vn)vn. 2 2 iii) For every v = a1v + ··· + anvn ∈ V one has: ||v|| = |a1| + ··· + |an| . 6) True of false (justify if not obvious): a) If A, B ∈ F n×n are non-zero and self-adjoint, so are 2A − 3B, A − (1 + i)B, xA + yB for all x, y ∈ F . b) If A, B ∈ F n×n are non-zero and self-adjoint, so are A5, AB,(A ± B)2. c) If A, B are unitarily equivalent, i.e., B = SAS−1 for some unitary matrix S, then: i) A is unitary iff B is so. ii) A is self-adjoint iff B is so. iii) A is normal iff B is so. iv) A is (semi-)positive iff B is so. v) A is diagonalizable iff B is so. d) The same questions in the case A, B are orthogonally equivalent, i.e., B = SAS−1 for an orthogonal matrix S. (The answer might depend on whether F = R or F = C.) e) The same questions in the case A, B are abstractly equivalent, i.e., B = SAS−1 for some invertible matrix S. (The answer might depend on whether F = R or F = C.) 7) Let V and W be finite dimensional F -vector spaces with inner product, and T : W → V be an operator, and T ∗ : W → V be its adjoint operator. Prove the following assertions concerning the fundamental spaces: a) ker(T ) = (im(T ∗))⊥ and im(T ) = (ker(T ∗))⊥. b) ker(T ∗) = (im(T ))⊥ and im(T ∗) = (ker(T ))⊥. c) ker(T ∗T ) = ker(T ) and im(T ∗T ) = im(T ∗). d) If V = W and T is self-adjoint, then ker(T ) = (im(T ))⊥ and im(T ) = (ker(T ))⊥. 8) For each of the matrices A below find a representation of the form A = U∆ with U unitary and ∆ upper triangular with non-negative diagonal entries, respectively find a polar decomposition A = UN, i.e., with U unitary and N (semi)positive definite. 1 0 0 0 ! 1 i 0 0 1 0 1 1 0 a) A = b) A = 0 1 i c) A = 1 i 0 0 1 0 i 0 1 −1 0 0 1 2 9) Let V be a finite dimensional vector with inner product over C, and T : V → T self- adjoint. Prove the following: a) ||v + iT (v)|| = ||v − iT (v)|| for all v ∈ V . b) v + iT (v) = w + iT (w) if and only if v = w. c) I + iT and I − iT are isomorphisms, hence (I + iT )−1 and (I − iT )−1 exist. d) U := (I − iT )(I + iT )−1 is a unitary operator. Terminology: The above operator U is called the Cayley transform of T . Notice that the −1 1−ti map T 7→ U := (I − iT )(I + iT ) is similar to the map t 7→ z := 1+ti from R into the unit 1−ti circle (|z| = 1) in C.[Note that one has indeed 1+ti = 1 for all t ∈ R (WHY).] Google the term “Cayley transform” and learn more about it. 10) Answer/prove the following: a) Let T : V → V be both positive and unitary. Then T = I. b) Let T1, T2 : V → V be normal operators which commute, i.e., T1T2 = T2T1. Then T1 ± T2 and T1T2 are normal operators. 11) Let A ∈ F n×n with F = R or F = C be given. Prove/disprove the following assertions: ∗ a) In + AA is invertible. b) If A is self-adjoint, then In + A is invertible. c) If A is unitary, then A + cIn is invertible for 0 < c < 1. n×n n×n d) If A ∈ R , then B := A + iIn ∈ C is invertible. 12)∗ Let V be an R-vector space with inner product. Prove/disprove: a) The unitary operators T : V → V are precisely the orthogonal ones (WHY). b) Let T : V → V is unitary and T 2 = −I. Then the following hold: α) T ∗ = −T . β) dim(V ) is even, say dim(V ) = 2n, and there exists a subspace E ⊂ V such that dim(E) = n and the following are satisfied: - T (E) = E⊥ and T (E⊥) = E. ⊥ - If E = (v1, ... , vn) is an orthonormal basis of E, then E := (T v1, ... , T vn) is an orthonormal basis for E⊥ and so is A := (E, E ⊥) for V . ∗! 0m U m×m -[T ]A = with U ∈ R unitary. U 0m 3.