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I have a circulant correlation matrix that has only positive entries. (Because it is a correlation matrix, it is symmetric with diagonal entries of 1.) I am wondering about the entries of the square root matrix of this circulant correlation matrix has only positive entries. I tried with a bunch of autoregressive correlation matrices and am able to show the result. However, I was wondering if such a result exists, or if even a restrictive version exists and how one goes about proving this.

Here is my R code for my experiments:

circulant.1d.corr <- function(n, rho)
    #creates 1d AR sequence of length n (periodic in length)
{
    ff <- c(1, rho^(1:(n/2)))
    ff <- c(ff, rev(ff[2:((n + 1)/2)]))
    ff
}
make.circ.mat <- function(n, rho) {
    x <- circulant.1d.corr(n, rho)
    R <- NULL
    for (i in 1:n) {
        if (i > 1) x <- c(x[n], x[-n])
        R <- rbind(R, x)
     }
    R
}

Rhalf.circ <- function(n, rho) {
    R <- make.circ.mat(n, rho)
    R.eigen <- eigen(R, symmetric = T)
    R.eigen$vectors %*% diag(sqrt(R.eigen$values)) %*% t(R.eigen$vectors)
}

> Rhalf.circ(10, 0.05)[1,]
 [1]  9.993743e-01  2.499218e-02  9.373043e-04  3.905640e-05  1.708390e-06
 [6] -2.359698e-09  1.708390e-06  3.905640e-05  9.373043e-04  2.499218e-02
> Rhalf.circ(10, 0.1)[1,]
 [1]  9.974890e-01  4.993711e-02  3.746853e-03  3.123054e-04  2.730579e-05
 [6] -6.763812e-08  2.730579e-05  3.123054e-04  3.746853e-03  4.993711e-02
> Rhalf.circ(10, 0.25)[1,]
 [1]  9.839308e-01  1.239837e-01  2.331028e-02  4.864143e-03  1.059005e-03
 [6] -1.104196e-06  1.059005e-03  4.864143e-03  2.331028e-02  1.239837e-01
> Rhalf.circ(10, 0.5)[1,]
 [1] 0.9293938767 0.2407913830 0.0915253610 0.0384984646 0.0165463712
 [6] 0.0006556758 0.0165463712 0.0384984646 0.0915253610 0.2407913830
> Rhalf.circ(10, 0.75)[1,]
 [1] 0.80511919 0.33654932 0.19870997 0.12794897 0.08148659 0.01608999
 [7] 0.08148659 0.12794897 0.19870997 0.33654932
> Rhalf.circ(10, 0.9)[1,]
 [1] 0.64488327 0.36788703 0.27640050 0.21993968 0.17083677 0.07437758
 [7] 0.17083677 0.21993968 0.27640050 0.36788703
> Rhalf.circ(10, 0.95)[1,]
 [1] 0.5505803 0.3657788 0.3022261 0.2598229 0.2183690 0.1273030 0.2183690
 [8] 0.2598229 0.3022261 0.3657788
>

Then, but for very small values, the entries of the square root matrix are positive.

I am calculating the square root of a matrix through its spectral decomposition.

Any pointers on the existence of this result (even a restricted version, such as decreasing AR(1) structure, which is the example in the R code above) or how to go about proving this is greatly appreciated.

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