Abstract: This paper discusses how to reduce A₀ to Hessenberg matrix by using Gauss elimination method of partial principal elements using elementary matrix technology.In order to make the numerical stability,the essential basic problem of how to exchange is emphasized.The first part briefly summarizes the matrix formula of reduction method.The second part further clarifies the basis of deducing the formula form of recursive reduction operation rule.The third part focuses on the details of recursive algorithm complete steps and logic implementation of the reduction method,and clearly states the fact that the final reduction result is consistent with the accurate calculation result of the matrix formula.The fourth part is a concrete example to verify the conclusion:the reduction method is based on sufficient calculation basis and is actually compact and feasible.

Key words: Element replacement, Elementary matrix, Exchange of ranks, Matrix block, Recursive reduction, Reduced matrix