EXERCISES . Suppose the LU decomposition of a matrix A is given by L= 1 1 0 and

Question

EXERCISES . Suppose the LU decomposition of a matrix A is given by L= 1 1 0 and = 0 1 2 Using only L, and f, (that is, without explicitly determining A ). solve Ax = f, when f = (6,9, 10) ·Suppose that solving a general n by n linear system of the form Ax = f by Gauss elimination takes 10 seconds on a given computer if n = 1000 Estimate how much time it will take to solve a 1000 by 1000 system Lg = f, followed by solving Ux = g, where L is lower triangular with 1's along its main diagonal, and where U is upper triangular? Thus you may assume that L and U have already been computed 45

Explanation / Answer

Create a matrix of real numbers and compute its transpose. B has the same elements as A, but the rows of B are the columns of A and the columns of B are the rows of A.

A = magic(4)
A = 16 2 3 13 5 11 10 8 9 7 6 12 4 14 15 1
B = A.'
B = 16 5 9 4 2 11 7 14 3 10 6 15 13 8 12 1

Complex Matrix

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Create a matrix containing complex elements and compute its nonconjugate transpose. B contains the same elements as A, except the rows and columns are interchanged. The signs of the imaginary parts are unchanged.

A = [1 3 4-1i 2+2i; 0+1i 1-1i 5 6-1i]
A = 1.0000 + 0.0000i 3.0000 + 0.0000i 4.0000 - 1.0000i 2.0000 + 2.0000i 0.0000 + 1.0000i 1.0000 - 1.0000i 5.0000 + 0.0000i 6.0000 - 1.0000i
B = A.'
B = 1.0000 + 0.0000i 0.0000 + 1.0000i 3.0000 + 0.0000i 1.0000 - 1.0000i 4.0000 - 1.0000i 5.0000 + 0.0000i 2.0000 + 2.0000i 6.0000 - 1.0000i

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