介紹
![多項式矩陣](/img/7/95c/wZwpmLwITM4gjMzcDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzLwMzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/2f8/wZwpmLwIDNxAjMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/e/b02/wZwpmLxQDN4EDO0cDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3AzLwczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/2f8/wZwpmLwIDNxAjMxMTMzEDN0UTMyITNykTO0EDMwAjMwUzLzEzLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
多項式矩陣即元為多項式的矩陣。 為 矩陣,或多項式矩陣,其中 是 的多項式。多項式矩陣,也稱為λ-矩陣、矩陣係數多項式(不是矩陣多項式),是數學中矩陣論里的概念,指係數是多項式的方塊矩陣。使用“λ-矩陣”的名稱時,說明係數多項式以λ為不定元。
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/d/9e1/wZwpmLyYjN2gDM4YzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzL2UzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
若n階多項式矩陣 的行列式 (非零多項式),則稱 是滿秩的(秩=n)或非奇異的。
![多項式矩陣](/img/1/40a/wZwpmL1ADNxgjN3YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzLwQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/e/3eb/wZwpmLxATO4MTOxkjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzLyIzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/b/903/wZwpmL1czN0QDO1YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLzMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
若使 ,則稱 是可逆的,或稱 是單模矩陣,記為 。
多項式矩陣
運算註解
多項式矩陣的加法、數乘、及乘法與一般矩陣的運算規則一樣,只是在運算過程中將式的運算換成多項式的運算即可。
多項式矩陣也像數字矩陣那樣定義行列式,並且多項式矩陣行列式的性質與數字矩陣行列式的性質相同。
初等變換
①互換的任意兩行(列)
②以非零數c乘以矩陣的一行(列)
![多項式矩陣](/img/7/7ea/wZwpmLwETMzIzN0cjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzL0gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
③矩陣的某一行(列)乘以多項式 )後某一行加到另一行(列)
多項式矩陣的秩
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
如果多項式矩陣 有一個r階子式不為零,而所有的r+1階子式全為零,則稱 的秩為r,零矩陣的秩規定為0。
多項式矩陣的逆矩陣
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
設 為n階λ-矩陣,如果存在n階λ-矩陣 ,使 = =l,則稱 可逆,並稱 為 的逆矩陣。
多項式矩陣的等價
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/a/9f1/wZwpmLxQTO1UjMxQTM2EzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL1UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
設 、 ,如果有 經過有限次初等變換能變為 ,則稱多項式矩陣 稱為與 等價,記為 。
多項式矩陣的行列式因子
1、定義
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/9/104/wZwpmLxUjN0kjMyQTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzLwEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/aba/wZwpmL3EDMwgTOwkzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5czL2IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
設 矩陣的秩為r,對於正整數k, , 中必有非零的k級子式, 中全部k級子式的首項係數為1的最大公因式 稱為 的k階行列式因子。
![多項式矩陣](/img/1/174/wZwpmL1YjN1cTNwkzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5MzL3IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
2、不變因子
![多項式矩陣](/img/b/602/wZwpmLwQTM5gDN0QTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzL1MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
中所有因子稱為 的不變因子組。
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/675/wZwpmL0QTOxUzN5IDM3UzM1UTM1QDN5MjM5ADMwAjMwUzLyAzLyMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
經過初等變換不改變多項式矩陣的秩和行列式因子,有相同的行列式因子或不變因子是 與等價的充要條件。
3、初等因子
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/0/be5/wZwpmLyUTOxMzMykDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzL4UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
設矩陣的不變因子為的標準分解式是
![多項式矩陣](/img/b/853/wZwpmL4gjM5UTN3UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzLwczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![多項式矩陣](/img/f/58a/wZwpmLxYDM2kjM3UTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzLyUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/e/181/wZwpmL4YTNwIjN5YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzL4AzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
則稱的標準分解式中的一次因式的方冪為A的初等因子。A中所有的初等因子稱為A的初等因子組。由定義知,初等因子是由不變因子確定的。
多項式矩陣的標準型
1、Smith標準形
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
對任一非零多項式矩陣 ,都等價於下列形式的矩陣(經過矩陣的初等變換實現):
![多項式矩陣](/img/e/8d9/wZwpmLxATN4QzNygTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLwczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/c/46b/wZwpmL1ATN5IDO1MDN3UzM1UTM1QDN5MjM5ADMwAjMwUzLzQzL0czLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/7/246/wZwpmL0ATNxgzN5kjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![多項式矩陣](/img/b/60f/wZwpmLyQDM1MjN0QTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzL0gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![多項式矩陣](/img/3/392/wZwpmL4MTNxcjMzYzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzLzAzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![多項式矩陣](/img/f/58a/wZwpmLxYDM2kjM3UTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzLyUzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/c35/wZwpmL4EDN5cDM0cDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL3gzL0gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
則稱它是 的更密斯(Smith)標準形,其中是它的秩,是首項係數為1的多項式,且。其中主對角線上的非零元素稱為 的不變因子。
2、Jordan標準形
形如
![多項式矩陣](/img/6/b16/wZwpmLwczN0kDMwcjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzL2gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![多項式矩陣](/img/2/e1b/wZwpmL3czMyQDO2QTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL0czLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![多項式矩陣](/img/8/754/wZwpmL0ADO3gTNwAzNwMzM1UTM1QDN5MjM5ADMwAjMwUzLwczLzYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
的方陣稱為階Jordan塊,其中可以是實數,也可以是複數。
套用
1、矩陣理論在計算機方面的套用,如矩陣的奇異值分解的套用,QR分解在網路方面的套用,還有在三維圖形圖像方面的套用。
2、多項式矩陣理論在網路分析中的套用,基於迴路矩陣B、基本割集矩陣Q和支路伏安特矩陣,藉助多項式理論中有關解耦零點的概念和理論,研究網路的複雜度和穩定性。
3、多項式矩陣理論知識,在建立和完善線性控制系統理論過程中具有基礎作用,套用廣泛。