Exercise 12. Show that the function p(x) = (x,Bx) + (p, x), for a fixed matrix B and a fixed vect...
A function f : Rn λε [0,1] R is strictly convex if for all x, y є Rn and all fax + (1-λ)y) < λ/(x) + (1-1)f(y) A symmetric matrix P-AT +A is called positive-definite if all its eigenvalues are positive. Show that a quadratic function f(x) -xPx is a convex function if and only P is positive-definite.
A function f : Rn λε [0,1] R is strictly convex if for all x, y є Rn and all fax +...
linear optimization
Assume that f : D → R is twice continuously differentiable for all x D, where the domain D off is an open, convex subset of Rn. Sh ▽2f(x), is symmetric positive-semi-definite for all x E D if and only if f is a convex function on D Moreover, if its Hessian matrix. ▽2 (x), is symmetric positive-definite for all x E D, then f is a strictly convex function on D Show that the converse of this...
3.52 Let A be an mxm positive definite matrix and B be an mxm
nonnegative definite matrix.
3.51 Show mal Il A IS à nonnegative definite matrix and a 0 for some z, then ai,-G3 = 0 for all j definite matrix. (a) Use the spectral decomposition of A to show that 3.52 Let A be an m x m positive definite matrix and B be an m × m nonnegative with equality if and only if B (0). (b)...
Exercise 5.2. This problem concerns the formulation of a model-based method for min imizing a twice-continuously differentiable function f : Rn R. Let B be a symmetric positive-definite matrix. At a point xk, consider the quadratic model (a) Write the quadratic model in terms of the variables p -x - k and find pk such that PERn (b) Show that the vector pk of part (a) is a descent direction for f at xk (c) Show that if B is...
(f) Let A be symmetric square matrix of order n. Show that there exists an orthogonal matrix P such that PT AP is a diagonal matrix Hint : UseLO and Problem EK〗 (g) Let A be a square matrix and Rn × Rn → Rn is defined by: UCTION E AND MES FOR THE la(x, y) = хтАУ (i) Show that I is symmetric, ie, 14(x,y) = 1a(y, x), if a d Only if. A is symmetric (ii) Show that...
Find numbers A and B which are fixed and positive such that f(x) = Bx is a probability density function on [0; A] with a median of 3.
3.52 Let A be an mxm positive definite matrix and B be an mxm
nonnegative definite matrix.
3.51 Show mal Il A IS à nonnegative definite matrix and a 0 for some z, then ai,-G3 = 0 for all j definite matrix. (a) Use the spectral decomposition of A to show that 3.52 Let A be an m x m positive definite matrix and B be an m × m nonnegative with equality if and only if B (0). (b)...
A real symmetric matrix B e Rnxn (i.e. BT = B) is said to be positive definite if all of its eigenvalues 11, 12, ..., In are positive. (Recall that is an eigenvalue of B if and only if there exits a nonzero vector t such that Bt = it). Show that B-1 is also positive definite. That is, you need to show that all the eigenvalues of B-1 are also positive. (Hint: consider equation Bt; = liti for all...
Problem 8: (11 total points) Suppose that B is a nx n matrix of the form B = Viv] + v2v + V3v3, where V1, V2, V3 € R”, n > 3 are nonzero column vector and are orthogonal. a) Show that B is a positive semidefinite matrix. b) Under which condition, B will be a positive definite matrix? c) Let A be 3x3 real symmetric matrix with eigenvalues 11 > 12 > 13. Let F be a positive definite...
3. Show that (a) the function g: R” → R, given by g(x) = ||2||2, is convex. (b) if f : RM → R is convex, then g:R" + R given by g(x) = f(Ax – b) is also convex. A here is an m x n matrix, and b ERM is a vector. You may use any of the results we covered in class (but the definition of convexity may be an easy way to do this, and gives...