(a) Let A be a Hermitian matrix. DEFINE: A is positive definite. (b) Let A be an n × n Hermitian ...
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)...
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....
I need help with a, b, and c. 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...
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)...
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)...
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) 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"...
Question B 7. (a) Let -1 0 0 (i) Find a unitary matrix U such that M-UDU where D is a diagonal matrix. 10 marks] (i) Compute the Frobenius norm of M, i.e., where (A, B) = trace(B·A). [4 marks] 3 marks] (iii) What is NM-illp? (b) Let H be an n × n complex matrix (6) What does it mean to say that H is positive semidefinite. (il) Show that H is positive semidefinite and Hermitian if and only...
Let A be a Hermitian n x n matrix, that is A = A^*. Write a MATLAB code for computing the LDL^* factorization for A , where D is a diagonal matrix with real diagonal entries , and L is a unit lower triangular matrix.
Exercise 1.4.61 This Exercise generalizes Propositions 1.4.51 and 1.4.53. Let A be an nxn positive definite matrix, let ji, j2, ..., jk be integers such that 1 < <j2 <... ik <n, and let X be the k x k matrix obtained by intersecting rows j1, ...,jk with columns 11,...,jk. Prove that A is positive definite.