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

Timothy Bates

9 years ago

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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.

In R, matrix inversion (usually signified by A ^-1) is done using the solve() function.

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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