# correlation matrix test: is this code correct or is it missing a multiple comparisons correction?

I have $m$ variables $x_1,\dots,x_m$, measured in $N$ independent tests $\{x_{i1},\dots,x_{im}\}_{i=1}^N$, leading to the design matrix $X$. I noted that the demo function corrplot_intro.r from the R package corrplot includes a nice function cor.mtest (reported below), which computes the pairwise correlations for $x_1,\dots,x_m$ in the sample $X$ and reports the corresponding $p$-values:

cor.mtest <- function(mat, conf.level = 0.95){
mat <- as.matrix(mat)
n <- ncol(mat)
p.mat <- lowCI.mat <- uppCI.mat <- matrix(NA, n, n)
diag(p.mat) <- 0
diag(lowCI.mat) <- diag(uppCI.mat) <- 1
for(i in 1:(n-1)){
for(j in (i+1):n){
tmp <- cor.test(mat[,i], mat[,j], conf.level = conf.level)
p.mat[i,j] <- p.mat[j,i] <- tmp$p.value lowCI.mat[i,j] <- lowCI.mat[j,i] <- tmp$conf.int[1]
uppCI.mat[i,j] <- uppCI.mat[j,i] <- tmp$conf.int[2] } } return(list(p.mat, lowCI.mat, uppCI.mat)) }  However, it seems to me that no correction for multiple comparisons is performed, while it would be needed because I'm (blindly) testing for all correlations (see this nice answer). So, the$p$-values reported are not actually reliable. Am I right? Could you help me correct the code so that the effect of doing$m^2\$ comparisons is correctly taken into account?