how do i calculate correlation between corresponding columns of two matrices and not getting other correlations as output

The first answer above calculates all pairwise correlations, which is fine unless the matrices are large, and the second one doesn't work. As far as I can tell, efficient computation must be done directly, such as this code borrowed from borrowed from the arrayMagic Bioconductor package, works efficiently for large matrices:

> colCors = function(x, y) { 
+   sqr = function(x) x*x
+   if(!is.matrix(x)||!is.matrix(y)||any(dim(x)!=dim(y)))
+     stop("Please supply two matrices of equal size.")
+   x   = sweep(x, 2, colMeans(x))
+   y   = sweep(y, 2, colMeans(y))
+   cor = colSums(x*y) /  sqrt(colSums(sqr(x))*colSums(sqr(y)))
+   return(cor)
+ }

> set.seed(1)
> a=matrix(rnorm(15),nrow=5)
> b=matrix(rnorm(15),nrow=5)
> diag(cor(a,b))
[1]  0.2491625 -0.5313192  0.5594564
> mapply(cor,a,b)
 [1] NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
> colCors(a,b)
[1]  0.2491625 -0.5313192  0.5594564

I would probably personally just use diag:

> diag(cor(a,b))
[1]  1.0000000 -1.0000000 -0.6964286

But you could also use mapply:

> mapply(cor,a,b)
         a          b          c 
 1.0000000 -1.0000000 -0.6964286