Let U be any m × m orthogonal matrix , and so by definition UTU = I. Let L′ = LUT and Y′ = UY. Then L′ is a (k × m) × (m × m) = k × m matrix and Y′ is a (m × m) × (m × 1) = m × 1 column vector. Also
X = μ + LY+ ε = μ + LUTUY + ε = μ + L′Y′ + ε
E[Y′] = E[UY] = U E[Y] = U0 = 0
var(Y′) = var(UY) = U var(Y) UT = UIUT = UUT = I
cov(Y′, ε) = cov(UY, ε) = U cov(Y, ε) = U0 = 0
This shows that if L and Y satisfy the model, then so do L′ and Y′. Since there are an infinite number of orthogonal matrices U, there are an infinite number of alternative models.
A rotation of the original axes is determined by an orthogonal matrix U with det = 1 (Property 6 of Orthogonal Vectors and Matrices). Thus, replacing Y and by Y′ is equivalent to rotating the axes. This won’t change the overall variance explained by the model (i.e. the communalities), but it will change the distribution of variances among the factors.
We seek an m × m rotation matrix U = [uij] such that the rows represent the existing factors and the columns represent the new factors. The most popular rotation approach is called Varimax, which maximizes the differences between the loading factors while maintaining orthogonal axes. Varimax attempts to maximize the value of V where
We can carry out the Varimax orthogonal rotation in standard Excel as described in Varimax. Because the calculation is complicated and time consuming, we suggest that you use following supplemental array function.
Real Statistics Functions: The Real Statistics Resource Pack provides the following supplemental function where R1 is a k × m range in Excel.
VARIMAX(R1): Produces a k × m array containing the loading factor matrix after applying a Varimax rotation to the loading factor matrix contained in range R1.
Figure 1 – Loading factors after Varimax rotation
We now see that the each of the variables (including Motivation) with the single exception of Entertainment correlates highly with only one factor. Also note that the communalities (column M) are the same as those shown in Figure 2 of Determining the Number of Factors prior to rotation.
We can also calculate the rotation matrix U that transforms the matrix in Figure 2 of Determining the Number of Factors into that in Figure 1. We do this by using Gaussian elimination (see Determinants and Linear Equations). In order to avoid the time consuming steps required in standard Excel, we first create a copy of the original loading factors (from Figure 2 of Determining the Number of Factors) and put a copy of the rotated loading factors (from Figure 2) right next to it as shown in Figure 3.
Figure 2 – Preparation for Gaussian elimination
We next apply the supplemental Excel function =ELIM(A57:H65) to get the result shown in Figure 3.
Figure 3 – Rotation Matrix
The 4 × 4 rotation matrix U is now found in the upper right portion (range E67:H70) of Figure 3. Note too that U is an orthogonal matrix (i.e. UTU = I) and det(U) = 1.