Degrees of freedom for the residuals of a principal component analysis — A clarification

Various conflicting proposals for degrees of freedom associated with the residuals of a principal component analysis have been published in the chemometrics-oriented literature. Here, a detailed derivation is given of the ‘standard’ formula from statistics. This derivation intends to be more accessible to chemometricians than, for example, the impeccable, but condensed proof that was published by John Mandel in a relatively unknown paper (J. Res. Nat. Bur. Stand., 74B (1970) 149–154). The derivation is presented in the form of a two-stage recipe that also appears to apply to more complex multiway models like the ones considered by Ceulemans and Kiers (Br. J. Math. Stat. Psych., 59 (2006) 133–150).