Choleski upper triangular decomposition

[Martyn Colins <martyncollins10@yahoo.com>]

 

I would like to obtain an upper triangular decomposition of a correlation matrix where all the diagonal terms are strictly above zero. The correlation matrix that I have is of full rank and has a non-zero determinant.

 

The matrix is defined as follows:

 

my.matrix <- matrix(0, 16, 16)

my.matrix[lower.tri(my.matrix)] <- c(

0.68, 0.79, 0.79, 0.83, 0.85, 0.86, 0.85, 0.85, 0.84, 0.82, 0.85, 0.82, 0.76, 0.81, 0.85,

0.82, 0.76, 0.66, 0.74, 0.65, 0.60, 0.51, 0.61, 0.70, 0.60, 0.69, 0.82, 0.63, 0.79,

0.74, 0.70, 0.75, 0.68, 0.64, 0.56, 0.66, 0.71, 0.62, 0.70, 0.77, 0.66, 0.73,

0.72, 0.78, 0.69, 0.65, 0.56, 0.69, 0.72, 0.62, 0.75, 0.75, 0.69, 0.75,

0.80, 0.73, 0.71, 0.62, 0.74, 0.73, 0.65, 0.75, 0.69, 0.72, 0.72,

0.79, 0.75, 0.72, 0.81, 0.79, 0.72, 0.83, 0.75, 0.75, 0.81,

0.70, 0.71, 0.74, 0.71, 0.70, 0.71, 0.67, 0.68, 0.75,

0.66, 0.71, 0.67, 0.66, 0.69, 0.64, 0.66, 0.72,

0.71, 0.62, 0.74, 0.61, 0.53, 0.57, 0.74,

0.72, 0.69, 0.75, 0.65, 0.70, 0.73,

0.65, 0.74, 0.70, 0.69, 0.74,

0.63, 0.60, 0.61, 0.73,

0.72, 0.73, 0.75,

0.67, 0.74,

0.69)

my.matrix <- my.matrix + t(my.matrix)

diag(my.matrix) <- 1

 

It has full rank:

qr(my.matrix)$rank

16

 

I get the following warning message:

decomp <- chol(my.matrix)

 

Warning messages:

Choleski decomposition not of full rank in: chol(my.matrix)

 

So I tried pivoting the choleski decomposition

decomp <- chol(my.matrix, pivot=T)

pivot <- attr(decomp, "pivot")

oo <- order(pivot)

t(decomp[, oo]) %*% decomp[, oo] # recover my.matrix

 

This does provide a decomposed matrix of full rank

qr(decomp)$rank

16

 

However, the diagonal elements are not all above zero:

 

decomp[row(decomp[,oo])==col(decomp[,oo])]

[1] 1.0000000 0.7332121 0.5763958 0.5394622 0.5369689 0.5339106 0.5294306 0.5174887 0.5167139 0.4963401 0.4905325 0.4557844

[13] 0.4095299 0.3783933 0.2495500 -0.6735201

 

(last element is not above zero).

 

Question is: how do I get an upper triangular decomposition of my original correlation matrix where all the diagonal terms are above zero?

 

Many thanks for any suggestions.

Regards

Martyn

 

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