Basic matrix algebra in SVD

Question

For my math course I'm reading about SVD in Principal Component Analysis (Abdi et al., 2010), I get stuck in (I think) a simple matrix algebra. In the text, equation (1) define SVD as:
$X = PΔQ^T$ (Eq.1)
and factor scores $F$ is defined as:
$F = PΔ$ (Eq. 2)
The matrix $Q$ gives the coefficients of the linear combinations used to compute the factors scores. This matrix can also be interpreted as a projection matrix because multiplying X by $Q$ gives the values of the projections of the observations on the principal components. This can be shown by combining Eqs. 1 and 2 as:
$F = PΔ = PΔQ^TQ = XQ$
I need help to see how to combining Eq1 and 2 I get $F=XQ$.
Thanks!

Answer 1

The trick is that Q is an orthogonal matrix ($QQ^T = Q^TQ = I$). Since we have $X = P \Delta Q^T $ and $ F = P \Delta $. Then by substitution: $$X = F Q^T $$ $$X Q= F Q^T Q$$ Then $$X Q = F $$

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