The SOR method of iteration has an iteration matrix G given by where w is a real number, L is str...
1 point) Find the LU factorization of 4 -5 -20 23 That is, write A LU where L is a lower triangular matrix with ones on the diagonal, and U is an upper triangular matrix A=
1 point) Find the LU factorization of 4 -5 -20 23 That is, write A LU where L is a lower triangular matrix with ones on the diagonal, and U is an upper triangular matrix A=
(1 point) Find the LU factorization of -g 3 -3 A = 4 LU where L is a lower triangular matrix with ones on the diagonal, and U is an upper triangular matrix. That is, write A A =
5. A 3 × 3 matrix is given by A=1020 -i 0 1 (a) Verify that A is hermitian. (b) Calculate Tr (A) and det (A), where det (A) represents the determinant of A (c) Find the eigenvalues of A. Check that their product and sum are consistent with Prob. (5b) (d) Write down the diagonalized version of A (e) Find the three orthonormal eigenvectors of A. (f) Construct the unitary matrix U that diagonalizes A, and show explicitly that...
(911 (1) (a) Recall that a square matrix A has an LU decomposition if we can write it as the product A = LU of a lower triangular matrix and an upper triangular matrix. Show that the matrix 0 1 21 A= 3 4 5 (6 7 9] does not have an LU decomposition 0 0 Uji U12 U13 O 1 2 Il 21 l22 0 0 U22 U23 = 3 4 5 (131 132 133 0 0 U33 6...
2. (a) Let A be the matrix A -4 21 8 -40 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P. Use Gaussian elimination with partial pivoting to find an upper triangular matrix U, permutation matrices Pi and P2 and lower triangular matrices Mi and M2 of the form 1 0 0 Mi-1A1 10 a2 0 1 M2 0 0 0 b1 with ail...
3. Given the matrix [ -1 2 -1] A= 3 2 1 10 10 1 Following steps (a)(b) to obtain the LU decomposition of the matrix A with partial piv- oting (a) Apply the Gaussian elimination method with partial pivoting to obtain an upper trian- gular matrix U. Record the corresponding permutation matrix for each pivoting step, and the numbers lik used to eliminate the zeros in column k. (b) Based on (a), express the matrices P, L and U...
ALTSIS AND NUMERICAL ANALYSIS 2. (a) Let A be the matrix 2 -115 8-4 Write down the 3 x 3 permutation matrix P such that PA interchanges the 1st and 3rd rows of A. Find the inverse of P Use Gaussian elimination with partial pivoting to find an upper triangular matix U, permutation matrices Pi and P2 and lower triangular matrices M and M2 of the form 1 0 0 0 1 1 0 0 0 bi 1 with land...
Differention Equations - Can someone answer the checked
numbers please?
Determinants 659 is the characteristic equation of A with λ replaced by /L we can multiply by A-1 to get o get Now solve for A1, noting that ao- det A0 The matrix A-0 22 has characteristic equation 0 0 2 2-A)P-8-12A +62- 0, so 8A1-12+6A -A, r 8A1-12 Hence we need only divide by 8 after computing 6A+. 23 1 4 12 10 4 -64 EXERCISES 1. Find AB,...
only do (e)-(g) The linear operator L : R3 + R3 is given by its matrix A = Al,s wit respect to the standard basis S = {(1, 22, 23}, where To 0 11 -10- 20 [4 00 (a) Find the characteristic polynomial PL(x) of L; (b) What are the eigenvalues of L and what are their algebraic multiplicities? (e) What are the geometric multiplicities of eigenvalues of L? Is L diagonal- izable? (d) Find a basis B of eigenvectors...
8 and 11
Will h x n lower triangular matrices. Show it's a w It's a 8. Dan will represent the set of all n x n diagonal matrices. Show it's a subspace of Mr. 9. For a square matrix AE M , define the trace of A, written tr(A) to be the sum of the diagonal entries of A (i.e. if A= a) then tr(A) = 211 + a2 + ... + ann). Show that the following subset of...