On The Symmetrical Property Of Procrustes Measure Of Distance

Abstract

~Icasurement the degree of differe11c:e bctwl'C'n two matric:<>s by using Procrustes analysis is preceded by a series of Euclidean si 111ilarit.v transformations namely translation, rotation. and dilation, performed i11 rc:;pcctccl order. for gai11i11g maximal matching. It b easy. by a counter cxainplc. to show that Procrustes measure do<.~ not obey the symmetrical property, something should he owned by any distnne<.' function. In thi:-. paper we analytically proved that normalization uvc>r co11figuratio11 mat ri<"es as au ct<l<lit io11al t ra11sfonnatio11 r<'::.ults in the satisfaction of the sy111metrical property hy Procrustc•:-. a11alysis. \Ve also proved that normalizatio11 can be undertaken prior to or aftl'r rotatio11 to preserve symmetrical property. ).forc'Ovcr, we proved that Procru:-.t<'s measure can be expressed in term of sint:,'Ular value.'> of the matrix. We here very much exploited the characteristic of full ~i11i.,'1 1l ar ,·al11<' clcco111po~itio11 under similarity trausformations.