1.6 Taking the inverse of a square matrix
The inverse of a matrix is denoted \(\mathbf{A}^{-1}\). You can think of the inverse of a matrix like \(1/a\). \(1/a \times a = 1\). \(\mathbf{A}^{-1}\mathbf{A} = \mathbf{A}\mathbf{A}^{-1} = \mathbf{I}\). The inverse of a matrix does not always exist; for one it has to be square. We’ll be using inverses for variance-covariance matrices and by definition (of a variance-covariance matrix), the inverse of those exist. In R, there are a couple way common ways to take the inverse of a variance-covariance matrix (or something with the same properties). solve() is the most common probably:
A = diag(3, 3) + matrix(1, 3, 3)
invA = solve(A)
invA %*% A [,1] [,2] [,3]
[1,] 1.000000e+00 -6.938894e-18 0
[2,] 2.081668e-17 1.000000e+00 0
[3,] 0.000000e+00 0.000000e+00 1A %*% invA [,1] [,2] [,3]
[1,] 1.000000e+00 -6.938894e-18 0
[2,] 2.081668e-17 1.000000e+00 0
[3,] 0.000000e+00 0.000000e+00 1Another option is to use chol2inv() which uses a Cholesky decomposition:
A = diag(3, 3) + matrix(1, 3, 3)
invA = chol2inv(chol(A))
invA %*% A [,1] [,2] [,3]
[1,] 1.000000e+00 6.938894e-17 0.000000e+00
[2,] 2.081668e-17 1.000000e+00 -2.775558e-17
[3,] -5.551115e-17 0.000000e+00 1.000000e+00A %*% invA [,1] [,2] [,3]
[1,] 1.000000e+00 2.081668e-17 -5.551115e-17
[2,] 6.938894e-17 1.000000e+00 0.000000e+00
[3,] 0.000000e+00 -2.775558e-17 1.000000e+00For the purpose of this course, solve() is fine.