3 Corrected typo to "A(2,3) = A(3,2)" edited Jun 19 '15 at 16:36 Mark L. Stone 11.1k11 gold badge1919 silver badges4747 bronze badges The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(12,3) = A(3,12), A(2,4) = A(4,2), A(3,4) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000  The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(1,3) = A(3,1), A(2,4) = A(4,2), A(3,4) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000  The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(2,3) = A(3,2), A(2,4) = A(4,2), A(3,4) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000  2 Corrected typo: Now reads "A(3,4) = A(4,3)". edited Jun 19 '15 at 14:25 Mark L. Stone 11.1k11 gold badge1919 silver badges4747 bronze badges The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(1,3) = A(3,1), A(2,4) = A(4,2), A(4,3,4) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000  The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(1,3) = A(3,1), A(2,4) = A(4,2), A(4,3) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000  The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(1,3) = A(3,1), A(2,4) = A(4,2), A(3,4) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000  1 answered Jun 19 '15 at 13:19 Mark L. Stone 11.1k11 gold badge1919 silver badges4747 bronze badges The missing entries are not uniquely determined, unless additional information is known. The only requirements on the missing entries are that they be symmetric, i.e., A(1,3) = A(3,1), A(2,4) = A(4,2), A(4,3) = A(4,3), and that A be positive semi-definite. I used CVX to maximize the sum of the unknown entries, then separately ran it to minimize the sum of the unknown entries, then separately ran it to minimize the sum of absolute values of the unknown entries. This produces three different solutions, among an infinite number of possibilities. Maximum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.9928 0.9685 0.6000 0.9928 1.0000 0.9913 0.7000 0.9685 0.9913 1.0000  Minimum sum:  A = 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 -0.1400 0.0400 0.6000 -0.1400 1.0000 0.2200 0.7000 0.0400 0.2200 1.0000  Minimum sum of absolute value: A= 1.0000 0.5000 0.6000 0.7000 0.5000 1.0000 0.0000 0.0000 0.6000 0.0000 1.0000 0.1354 0.7000 0.0000 0.1354 1.0000