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  1. K

    Construction of minimum norm solution matrix

    Homework Statement Consider the linear system of equations Ax = b b is in the range of A Given the SVD of a random matrix A; construct a full rank matrix B for which the solution: x = B^-1*b is the minimum norm solution. Also A is rank deficient by a known value and diagonalizable...
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    Condition number of a matrix

    Homework Statement For a system of equations Ax = b Let dA be a random perturbation of the matrix A The error in Which dA fullfills the equality norm(A^-1 (da) x) = norm(A^-1) norm(dA) norm(x) (The SVD of A is known) (b is a known vector) Homework Equations The Attempt...
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    Effect of orthonormal projection on rank

    Homework Statement Given rank(R) and a QR factorization A = QR, what is the rank(A) Homework Equations The Attempt at a Solution I want to know if multiplication by a full rank orthonormal matrix Q and an upper trapezoidal matrix R yields rank(R)=rank(Q*R)=rank(A) This is...
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    Oblique projector

    Homework Statement Find the oblique projector P so where range(P) = range(U) and range(I-P) = range(W) Homework Equations P^2-P = 0 range(I-P) = null(P) The Attempt at a Solution It seems that U and W are complementary subspaces. According to...
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    Orthogonal projectors (minimization and variational problem)

    Homework Statement S1 is in subspace of C^n. P unique orthogonal projector P : C^n -> S1, and x is in range of C^n. Show that the minimization problem: y in range of S1 so that: 2norm(x-y) = min 2norm(x-z) where z in range of S1 and variational problem: y in range of S1 so that...
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    Solution to system of linear equations in range of system matrix

    Homework Statement See image. a) and b) have been solved. The problem is c) Homework Equations The Attempt at a Solution I really have no idea where to begin. For the three systems given there are solutions x in range(A) for system 1 and 2 but not for 3. Therefore I have been...