You simulate three correlated standard normals with correlations r12 = 0.5, r13 = 0.3, r23 = 0.4 using the Cholesky factor L of the correlation matrix, with X = L Z. Compute L_33, and say what it means when a Cholesky step asks for the square root of a negative number.

You simulate three correlated standard normals with correlations r12 = 0.5, r13 = 0.3, r23 = 0.4 using the Cholesky factor L of the correlation matrix, with X = L Z. Compute L_33, and say what it means when a Cholesky step asks for the square root of a negative number.

Approach: Run the Cholesky recursion row by row on the 3 by 3 correlation matrix, using the fact that each diagonal entry is the residual of 1 after subtracting the squares already placed in that row.

0.9092. The Cholesky decomposition gives L_11 = 1, L_21 = 0.5, L_22 = sqrt(1 - 0.25) = 0.8660, L_31 = 0.3 and L_32 = (0.4 - 0.5*0.3)/0.8660 = 0.25/0.8660 = 0.2887. Then L_33 = sqrt(1 - 0.3^2 - 0.2887^2) = sqrt(1 - 0.09 - 0.0833) = sqrt(0.8267) = 0.9092. Writing X = L Z with Z independent standard normals gives Cov(X) = L L^T = C, which is why this is the standard route to correlated normals in a simulation. A negative argument under a square root means the input matrix is not positive definite, so no such factor exists and the stated correlations are not jointly attainable, which is exactly what the recursion detects in O(n^3) work.

Follow-up: If one correlation is estimated with error and the matrix fails Cholesky, how would you find the nearest valid correlation matrix?

Key concepts: cholesky decomposition, correlated normals, positive definite, simulation.