Our data vector b has been measured with some error. Let btrue be the true but unknown data, and let Axtrue = birue. (a) The columns of the matrix V = [v₁,...,Vn] form an orthonormal basis for n-dimensional space. Let's express the solution Xtrue to the least squares problem as Xtrue W1V1+...+wnvn. Determine a formula for w; (i = 1,...,n) in terms of U, btrue, and the singular values of A. (b) Justify the reasoning behind these two statements. 1 A(x-Xtrue)=b-birue -r means ||x-Xtrue || ≤(b-btrue - r||), on btrue =Axtrue means ||btrue || = ||Axtrue||≤||A||||Xtrue ||.

"onsdensiks Consider the m x n-matrix A and the vector be Rm that are given by A = [aij], 6 = [bi] where aij = (–1)++i(i – j), b; = (-1)' for i = 1,... m and j = 1,..· , n. %3D Note that aij is the entry of A at the i-th row and the j-th column. Consider the following condition on vectors iE R": (Condition1) Aa õ. Write and run a python Jupyter notebook using the PULP package, to check whether there is a vector iE R" that satisfies (Condition1), • when m =n = 10, and if it exists, find one. Submit your screenshots or the pdf file of your Jupyter Notebook; it should show both the codes and the results. [For credits, you must write a python Jupyter notebook to solve this problem.] Python Hint: Start with giving М-10 N=10 You may want to use Python lists combined with 'forloop'. For example, column=[j+1 for j in range(N)] row=[i+1 for i in range(M)] Then, one can define the matrix A and the vector 6, using Python dictionaries combined with 'forloop' as a= {i:{j:(-1)**(i+j)*(i-j) for j in…

The matrix [A] has a norm of ||A||=3.2 x 10° and the norm of the inverse of that matrix is ||A-4||=2.5 x 1012 If a solution is obtained for a system of equations [A]{x}={b} using this same matrix [A] and double precision computations, then O the solution is very sensitive to input error O the solution is insensitive to input error O small residuals indicate accurate answers O small residuals do not indicate accurate answers O the system is ill-conditioned

Let f ∈ C+ 2π with a zero of order 2p at z. Let r>p and m =  n/r. Then there exists a constant c > 0 independent of n such that for all nsufficiently large, all eigenvalues of the preconditioned matrix C−1 n (Km,2r ∗ f)Tn(f) are larger than c.

2. For a n-vector x, and X1 + x2 X2 + x3 y = Ax = Xn-1 + Xn a) Find A b) Are the columns of A linearly independent? Justify your answer? c) Are the rows of A linearly independent? Justify your answer?

Assume A is k x n-matrix and P is k × k-invertible matrix. Prove that rank(PA) = rank(A).

For the matrix A, find (if possible) a nonsingular matrix P such that p-lAP is diagonal. (If not possible, enter IMPOSSIBLE.) 6 -3 A = -2 P = Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP =

How to solved with explanations based on mathematics computer graphics programming knowledge?

Perform the Singular Value Decomposition (SVD) of the complex matrix A into A = UΣV# with U and V being unitary matrices! i where = √-1 i 0 A = (1+2i) (1-i√2) 1 The goal is to find the SVD of the given complex matrix A.

use MATLAB only