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積和式及其計(jì)算
摘要
為了更好地用矩陣來(lái)描述組合問(wèn)題，我們引入1個(gè)矩陣置換相抵下的不變量——積和式。積和式的概念在1812年由Binet和Cauchy提出的。積和式是矩陣的1個(gè)重要參數(shù)，有深刻的組合意義，在組合理論中經(jīng)常將積和式與其他參數(shù)建立聯(lián)系，它類似于矩陣的行列式，但又有很大的區(qū)別。
本文給出了積和式的定義如下：設(shè) 是 × 矩陣（ ），則稱和式 為 的積和式（permanent）,這里 表示｛ ｝中所有 元排列的集合。
本文中詳細(xì)闡述了積和式、 矩陣積和式的1些性質(zhì)。在積和式的計(jì)算方面，闡述了利用Ryser定理計(jì)算積和式 的傳統(tǒng)方法；利用正行列式得到兩類 矩陣積和式，并給出其兩種類型的組合應(yīng)用，其后，利用正行列式建立了計(jì)算積和式 的另1種計(jì)算理論；最后還給出了關(guān)于雙隨機(jī)矩陣的兩個(gè)問(wèn)題的計(jì)算證明。
關(guān)鍵詞:積和式；Ryser定理； 矩陣；雙隨機(jī)矩陣；應(yīng)用
Abstract
In order to describe the question of combination with matrix better, We introduce a constant in replacement and balance out of matrix—— Permanent. The concept of permanent set up by Binet and Cauchy in 1812. It is an important parameter of matrix with profound significance of combination. It often connects permanent with other parameters in theory of combination. It is similar to the determinant of matrix, but there are very great differences.
Define the permanent as follows: It is supposed that is × matrix（ ）,so claim the permanent as the permanent of , Here is all —Permutation of｛ ｝.
The text described some properties of permanent、 matrix permanent 。At calculation for permanent, it described the tradition method of utilization Ryser theorem to calculate permanent ，Utilize the positive determinant to receive two kinds of matrix permanent, Provide its two types association application; Thereafter, it set up another kind of calculation theory of Calculation permanent that still utilize the positive determinant; finally, provide the identifications of two questions about bistochastic matrix.
Keywords:Permanent； -matrix；Ryser theorem；bistochastic matrix； Application.
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