7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Giv...
a through e is considered one question.
7.Let A be ann x n real symmetric invertible matrix, let B Rt and C E R. Define f:R R by 2 a. Give f (a) c. Give f"(x) d. Prove that if A is positive definite and u is the critical point of f, then f(u) < f(x) for all x E Rn where x Prove that if A is negative definite and u is the critical point of f, then f(u)...
Let A E(R") be Hermitian and positive definite, let v Define g R" R by R", and let cE R (a) Show that g is polynomial function of (... ,En) and in particular it has continuous partial derivatives of all orders. (b) Show that oo. Hint: Use Ezercise Ic. (c) Prove that g(x) achieves a global minimum d) Compute Vg(x). Show that g has a unique critical point, and hence argue that the minimum must be achieved at this point....
*14. Let A be an n x n matrix. Define f:R" R by f(x) = Ax.x = x'AX. (a) Show that f is differentiable and Df (a)h = Aah + Ah a. (b) Deduce that when A is symmetric, Df(a)h = 2Aa . h. 15. Let a € R", 8 >0, and suppose f: B(a, 8) - R is differentiable at a. Suppose f(a) f(x)
Let A EL(R") be Hermitian and positive definite, let vE R, and let cE R Define g R" R by (a) Show that g is polynomial function of (..,In) and in particular it has continuous partial derivatives of all orders. (b) Show that oo. Hint: Use Ezercise Ic. (c) Prove that g(x) achieves a global minimum (d) Compute ▽g(x). Show that g has a unique critical point, and hence argue that the minimum must be achieved at this point. (e)...
2. Let A be an n x n real symmetric matrix or a complex normal matrix. Prove that tr(A) = X1 + ... + and tr(AⓇA) = 1212 + ... +14.12 where ....... An are the eigenvalues of A repeated with multiplicity (for example, if n = 3 and the eigenvalues of A are -3 and 7 but -3 has multiplicity 2 then 11 = -3, 12 = -3, and Az = 7). 3. Let A be an n x...
Problem 4. Let A, B e Rmxn. We say that A is equivalent to B if there exist an invertible m x m n x n matrix Q such that PAQ = B. matrix P and an invertible (a) Prove that the relation "A is equivalent to B" is reflexive, symmetric, and transitive; i.e., prove that: (i) for all A E Rmx", A is equivalent to A; (ii) for all A, B e Rmxn, if A is equivalent to B...
(6) Let fel ), where is Lebesgue measure on R. Define F:R → R by F(x) = f' f(t) dx. (a) Prove that F is a continuous function. (b) Prove that F is uniformly continuous on R. (Note that R is not compact.)
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian matrix. PROVE: If A is positive definite the n every eigenvalue of A is positiv e. (c) Let Abe an n X n Hermitian matrix. PROVE: If every eigenvalue of A is positive. Then A is positive definite.
(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian...
(a) Let S be a symmetric positive definite matrix and define a function | on R" by 1/2 xx Sx . Prove that this function defines a vector norm. Hint: Use the Cholesky decomposition. (b) Find an example of square matrices A an This shows that ρ(A) is not a norm. Note: there are very simple examples. d B such that ρ(A+B)>ρ(A) + ρ(8)
(a) Let S be a symmetric positive definite matrix and define a function | on R"...