# What’s that “pre- and post-multiply” stuff?

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Often in SEM scripts you will see matrices being pre- and post-multiplied by some other matrix. For instance, this figures in scripts computing the genetic correlation between variables. How does pre- and post-multiplying a variance/covariance matrix give us a correlation matrix? And what is it that we are multiplying this matrix by?

In general, a covariance matrix can be converted to a correlation matrix by pre- and post-multiplying by a diagonal matrix with 1/SD for each variable on the diagonal.

For the diagonal case, the inverse of a matrix is simply 1/x in each cell.

## Example with variance matrix A

A = matrix(nrow = 3, byrow = T,c( 1,0,0, 0,2,0, 0,0,3) ); solve(A) [,1] [,2] [,3] [1,] 1 0.0 0.00 [2,] 0 0.5 0.00 [3,] 0 0.0 0.33

A number times its inverse = 1. For Matrices

`solve(A) %*% A = Identity Matrix`

solve(A) %*% A # = I: The Standardized diagonal matrix [,1] [,2] [,3] [1,] 1 0 0 [2,] 0 1 0 [3,] 0 0 1

### An example with values (covariances) in the off-diagonals

A = matrix(nrow = 3, byrow = T, c( 1, .5, .9, .5, 2, .4, .9, .4, 4) ); I = matrix(nrow = 3, byrow = T, c( 1, 0, 0, 0, 1, 0, 0, 0, 1) ); varianceA = I * A # zero the off-diagonal (regular, NOT matrix multiplication) sdMatrix = sqrt(varianceA) # element sqrt to get SDs on diagonal: SD=sqrt(var) invSD = solve(sdMatrix) # 1/SD = inverse of sdMatrix invSD [,1] [,2] [,3] [1,] 1 0.00 0.0 [2,] 0 0.71 0.0 [3,] 0 0.00 0.5

Any number times its inverse = 1, so this sweeps covariances into correlations

corr = invSD %*% A %*% invSD # pre- and post- multiply by 1/SD [,1] [,2] [,3] [1,] 1.00 0.35 0.45 [2,] 0.35 1.00 0.14 [3,] 0.45 0.14 1.00

## Easy way of doing this in R

Using diag to grab the diagonal and make a new one, and capitalising on the fact that inv(X) = 1/x for a diagonal matrix

diag(1/sqrt(diag(A))) %&% A # The %&% is a shortcut to pre- and post-mul [,1] [,2] [,3] [1,] 1.00 0.35 0.45 [2,] 0.35 1.00 0.14 [3,] 0.45 0.14 1.00

## Even-easier built-in way

cov2cor(A) [,1] [,2] [,3] [1,] 1.00 0.35 0.45 [2,] 0.35 1.00 0.14 [3,] 0.45 0.14 1.00 Note: See also this followup post on getting a correlation matrix when you are starting with a lower-triangular Cholesky composition.

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