(This article was first published on

**Industrial Code Workshop**, and kindly contributed to R-bloggers)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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