1. Construct

a 9×2 matrix X with some missing values, such that SX computed using all

available data for the covariance or correlation matrix is not nonnegative

definite.

2. Consider an m×m,

symmetric nonsingular matrix, R, with 1s on the diagonal and with all

off-diagonal elements less than 1 in absolute value. If this matrix is positive

definite, it is a correlation matrix. Suppose, however, that some of the

eigenvalues are negative. Iman and Davenport (1982) describe a method of

adjusting the matrix to a near-by matrix that is positive definite. (See

Ronald L. Iman and James

»

1. Construct

a 9×2 matrix X with some missing values, such that SX computed using all

available data for the covariance or correlation matrix is not nonnegative

definite.

2. Consider an m×m,

symmetric nonsingular matrix, R, with 1s on the diagonal and with all

off-diagonal elements less than 1 in absolute value. If this matrix is positive

definite, it is a correlation matrix. Suppose, however, that some of the

eigenvalues are negative. Iman and Davenport (1982) describe a method of

adjusting the matrix to a near-by matrix that is positive definite. (See

Ronald L. Iman and James M. Davenport, 1982, An Iterative Algorithm to Produce

a Positive Definite Correlation Matrix from an Approximate Correlation

Matrix, Sandia Report SAND81-1376, Sandia National Laboratories, Albuquerque,

New Mexico.) For their method, they assumed the eigenvalues are unique, but

this is not necessary in the algorithm. Before beginning the algorithm, choose

a small positive quantity,, to use in the adjustments, set k = 0, and set R(k)

= R.

3. Form

R(k) from R_ by setting all diagonal elements to 1.

4. Set

k = k + 1, and go to step 1. (The algorithm iterates on k until

p = 0.)

Write a

program to implement this adjustment algorithm. Write your program to accept

any size matrix and a user-chosen value for. Test your program on the

correlation matrix from Exercise 9.14

»

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