In this work, we focus on designing a transformation from arrowheaded to tridiagonal matrices by using a novel method "Tridiagonal Matrix Enhanced Multivariance Products Representation (TMEMPR)". We have quite recently developed "Arrow-headed Enhanced Multivariance Products Representation for A Kernel" decomposition method which produces arrowheaded core matrices. However tridiagonal matrix forms are preferred in most scientific fields. "Arrowheaded Enhanced Multivariance Products Representation for a Kernel (AEMPRK)", decomposing a linear univariate integral operator and its kernel which can be expressed as a finite sum of binary products composed of univariate functions, was developed and improved by M. Demiralp and his research group. In principal, TMEMPR, can tridiagonalize any type matrix, so the arrowheaded ones, by using only identity matrix weights or some other matrix weights. We especially emphasize on weight issues here in this work and show certain very interesting reductive features of TMEMPR.