this problem explores the relationship between condition number and the volume of a random paral

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This problem explores the relationship between the condition number and the volume of a random parallelepiped in 4-dimensional space whose sides are unit vectors. Let a1, a2, a3, a4 be vectors which emanate from a given vertex of the parallelepiped and describe its sides. By a well-known result from geometry, the volume of the parallelepiped is then det([a1, a2, a3, a4]). Your program for this problem should do the following. (a) Generate 4 random column vectors of size 4-by-1, viewing them as the 4 columns of a 4-by-4 random matrix A = rand(4). (b) Normalize the columns A = A * diag(1./sqrt(sum(A.*A))). Provide some comments to explain what this operation does. (c) Compute the volume of the parallelepiped defined by the normalized vectors. Your program must use the LU factorization for computing the determinant (that is, do not use the command det(A) here). (d) Compute the condition number κ(A) with cond(A). (e) Repeat the process 1000 times and produce a scatter plot of det(A) versus 1/κ(A). What are the maximum and minimum values found for |det(A)|? What is the condition number corresponding to each of these values? How can you explain the extreme values of the condition number and the corresponding values of the determinant? This is a Numerical Analysis problem. Please use MATLAB to write a code.

          This problem explores the relationship between the condition number and the volume of a random parallelepiped in 4-dimensional space whose sides are unit vectors. Let a1, a2, a3, a4 be vectors which emanate from a given vertex of the parallelepiped and describe its sides. By a well-known result from geometry, the volume of the parallelepiped is then det([a1, a2, a3, a4]). Your program for this problem should do the following.
(a) Generate 4 random column vectors of size 4-by-1, viewing them as the 4 columns of a 4-by-4 random matrix A = rand(4).
(b) Normalize the columns A = A * diag(1./sqrt(sum(A.*A))).
Provide some comments to explain what this operation does.
(c) Compute the volume of the parallelepiped defined by the normalized vectors. Your program must use the LU factorization for computing the determinant (that is, do not use the command det(A) here).
(d) Compute the condition number κ(A) with cond(A).
(e) Repeat the process 1000 times and produce a scatter plot of det(A) versus 1/κ(A).
What are the maximum and minimum values found for |det(A)|? What is the condition number corresponding to each of these values? How can you explain the extreme values of the condition number and the corresponding values of the determinant?
This is a Numerical Analysis problem. Please use MATLAB to write a code.

Added by Mary A.

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