(a) Prove that if matrix is positive definite (iAx > 0 for any r 0), then the Jacobi method converges for the linear system Ar b. (a) Prove that if matrix is positive definite (iAx > 0 for...
2a. Given a linear system of equations A b with a symmetric positive definite matrix A ERIX4 which has eigenvalues 1, 1/4, 1/9,1/16. Consider the iterative method defined by r(k +1) = r(k)-w(Ax(k)-b). Can you choose w such that method is convergent? If so, what is the best possible w? 2b. Discuss the convergence of the Jacobi method for Ar-b with the tridiagonal matrix -1 3 Does the Jacobi method converge for this matrix? What is the convergence rate 2a....
Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed from the definition to show that if a-(a a in n, then ar Ha>0 .a, is a nonzero vector a 1s a nonzero vector (ii)--= Í xi +j-2 ax (111) manipulate a' Ha into the integral of a positive function. i+ J Hta11 2. Prove that for the (Hilbert) matrix is positive definite. i+j-1 i.j-1 Hnts: (Proceed from the definition to show that if a-(a...
(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...
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)...
2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab) 2. Suppose that A is an rn x n matrix and b є С". Prove that the linear system CSA, b) is consistent if and only if r(A) = r(Ab)
5. Recall that a symmetric matrix A is positive definite (SPD for short) if and only if T Ar > O for every nonzero vector 2. 5a. Find a 2-by-2 matrix A that (1) is symmetric, (2) is not singular, and (3) has all its elements greater than zero, but (1) is not SPD. Show a nonzero vector such that zAx < 0. 5b. Let B be a nonsingular matrix, of any size, not necessarily symmetric. Prove that the matrix...
(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"...
If Matrix A, r(A)=n, prove that r(AB)=r(B), for any B nxp, and show that for any invertible mxm matrix P, there exists Q mxn with full rank such that A=PQ
Relevant Information: 1" (20%) (Linear systems) Given a linear system C1 +33 2 One can convert it into an iterative formula x(n+1) TX(m) + c where X(n) = (a (n),X(n), a (n))t įs the approximated solution at the nth iteration, T3x3 is the iterative matrix and caxi is the vector associated with the correspondent iterative method. (a) (5 %) Compute the associated matrix T and vector c associated with Jacobi method. (b) (5 %) Compute (T) and determine if Jacobi...