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The bilinear transformation (BT) is used for vocal tract length normalization (VTLN) in speech recogniton systems. We prove two properties of the bilinear mapping that motivated the band-diagonal transform proposed in M. Afify and O. Siohan, (ldquoConstrained maximum likelihood linear regression for speaker adaptation,rdquo in Proc. ICSLP, Beijing, China, Oct. 2000.) This is in contrast to what is stated in M. Pitz and H. Ney, (ldquoVocal tract length normalization equals linear transformation in cepstral space,rdquo IEEE Transactions on Speech and Audio Processing, vol. 13, no. 5, pp 930-944, September 2005) that the transform of Afify and Siohan was motivated by empirical observations.