直積簡介
![直積](/img/a/ed1/wZwpmL4ITO2IzMyQTM5czN0UTMyITNykTO0EDMwAjMwUzL0EzLwEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/c/e70/wZwpmL1ITO3IDO3AjMyADN0UTMyITNykTO0EDMwAjMwUzLwIzLyczLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/d/a87/wZwpmLwQTMzgjNzUjMxMzM1UTM1QDN5MjM5ADMwAjMwUzL1IzL2QzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/8/35e/wZwpmLxUTM5ADM4kTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5UzL1IzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直積](/img/9/5c4/wZwpmL2ITN5cDMwIDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/9/5c4/wZwpmL2ITN5cDMwIDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/9/5c4/wZwpmL2ITN5cDMwIDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/9/5c4/wZwpmL2ITN5cDMwIDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
對於兩個集合 U和 V,由 U的元素 和V的元素 的有序組 構成的集合 稱為 U和 V的直積集,記作 。當 U、 V具有像群、環等那樣的代數結構時,如果對 定義同樣的代數結構,則稱 是 U和 V的直積當 U、V具有拓撲結構時,可以通過對 定義適當的拓撲結構來定義 U和 V的直積。因為直積的概念是根據它的每個因子具有的結構在直積集中定義同樣的結構而產生的。所以它在數學的各個分支中都會出現。最簡單的例子是,可以把平面看作直線和直線的直積。群的直積、拓撲空間的直積、度量空間的直積、賦范線性空間的直積等都是重要的直積的例子。
不同種類的直積
群的直積
![直積](/img/9/43e/wZwpmLzEDMxkDOzczN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3czLxgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/2/865/wZwpmL4ETN4czNyUzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1czLxYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/3/6d5/wZwpmLwcTN1cTM5QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzLxIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
設 是兩個群,在 上定義一個二元合成如下:對 ,令
![直積](/img/e/bca/wZwpmL3ADOxcDO3YjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL2YzL3YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/2/865/wZwpmL4ETN4czNyUzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1czLxYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/9/43e/wZwpmLzEDMxkDOzczN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3czLxgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/9/43e/wZwpmLzEDMxkDOzczN2UzM1UTM1QDN5MjM5ADMwAjMwUzL3czLxgzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
關於這一合成構成群,稱為群 的 直積或 直積群。如果採用加法記號,則稱為群 的真和。
群的直積在群論研究中占有很重要的地位:群的外直積提供了由已知群構造新群的方法,群的內直積分解可把研究群G的結構化為研究其若干子群的結構,例如,任一有限生成交換群必可分解為若干循環子群的直積。
拓撲空間的直積
![直積](/img/3/7c3/wZwpmLyUzNzEzM4QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzL1gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/3/f8b/wZwpmL0IjN5IDN2ITN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyUzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/e/e17/wZwpmLxYjMxEDO1gzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4czLwMzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/b/de1/wZwpmL1IDM5UjNwkDO0ATN0UTMyITNykTO0EDMwAjMwUzL5gzL2gzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/1/2e5/wZwpmLyYDMzgzMwMzMzIDN0UTMyITNykTO0EDMwAjMwUzLzMzL1IzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/0/e88/wZwpmL1EjNzETOwMjM4IDN0UTMyITNykTO0EDMwAjMwUzLzIzL2gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/3/7c3/wZwpmLyUzNzEzM4QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzL1gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/3/f8b/wZwpmL0IjN5IDN2ITN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyUzLxUzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/3/7c3/wZwpmLyUzNzEzM4QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzL1gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設 是兩個拓撲空間,則 上以{ 是 的開集, 是 的開集}為基的拓撲,稱為 的拓撲的積拓撲,而賦予這一拓撲的直積 ,稱為拓撲空間 的直積。
度量空間的直積
![直積](/img/e/a81/wZwpmL2EjMwgjM0QTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL4gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直積](/img/a/80f/wZwpmL1gDOyYjNzkzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL5czLzIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/e/a81/wZwpmL2EjMwgjM0QTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL4gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直積](/img/5/57d/wZwpmL3gTNyIjN5gzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4czL1QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設 是兩個度量空間, 分別是 上的距離,如果對 ,令
![直積](/img/9/c7a/wZwpmL3YDO3IDM2cTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL3UzL0MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/d/f06/wZwpmL3QDM3czN0IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLyEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/d/f06/wZwpmL3QDM3czN0IjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLyEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/e/a81/wZwpmL2EjMwgjM0QTMxMzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL4gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
則d是 上的一個距離函式。賦予這一距離的 ,稱為度量空間 的直積。
賦范線性空間的直積
![直積](/img/b/250/wZwpmL0MTM5QTO0QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzLyAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/4/b9e/wZwpmLzEzNygzNyUzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1czL0gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/b/250/wZwpmL0MTM5QTO0QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzLyAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/3/ae4/wZwpmLyITN4QTOxYjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL2YzL3IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
設 是兩個賦范線性空間, 分別表示 上的範數。如果對 ,令
![直積](/img/c/dbc/wZwpmL1cDMzADMyMjN0YzM1UTM1QDN5MjM5ADMwAjMwUzLzYzL4UzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/4/89b/wZwpmL2EzN5kjMxUTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直積](/img/8/3f2/wZwpmLwQjM3cTM1QzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0czL4IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/3/ae4/wZwpmLyITN4QTOxYjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL2YzL3IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/b/59f/wZwpmLwMTO3MDM2IDO0YzM1UTM1QDN5MjM5ADMwAjMwUzLygzLxczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
則 是 上的—個範數。又對 , 。令
![直積](/img/7/b01/wZwpmL1gDOyAjNzQzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0czLxIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/e/920/wZwpmLyEjMxgjN5gTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL4UzL2gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![直積](/img/4/89b/wZwpmL2EzN5kjMxUTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzLxczLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直積](/img/8/3f2/wZwpmLwQjM3cTM1QzN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0czL4IzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/b/250/wZwpmL0MTM5QTO0QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzLyAzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
則 又成為一個線性空間。連同方才定義的範數 , 成為一個賦范線性空間,稱為賦范線性空間 的直積。
矩陣的直積
![直積](/img/f/b16/wZwpmL2EDNxMTN2YjN0YzM1UTM1QDN5MjM5ADMwAjMwUzL2YzL4AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
設 ,稱如下的分塊矩陣
![直積](/img/9/53c/wZwpmLyQjMwAjM1MTN0YzM1UTM1QDN5MjM5ADMwAjMwUzLzUzLzAzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![直積](/img/3/afa/wZwpmL0MzMyIzMxUTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL1UzL1EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
為 A與 B的 直積( 張量積或 Kronecker 積),記為。
![直積](/img/f/8ee/wZwpmLzUDOwEzMyAzN0YzM1UTM1QDN5MjM5ADMwAjMwUzLwczL1EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/9/93f/wZwpmLwETO4IjMxMDO4EDN0UTMyITNykTO0EDMwAjMwUzLzgzLyczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/9/5b3/wZwpmLxIzM2gDM0cTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL3UzLwczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/8/d1e/wZwpmL4IDMwMTM3QTN0YzM1UTM1QDN5MjM5ADMwAjMwUzL0UzL3EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/f/8ee/wZwpmLzUDOwEzMyAzN0YzM1UTM1QDN5MjM5ADMwAjMwUzLwczL1EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/5/0a1/wZwpmL4QjN4MDO4czN0YzM1UTM1QDN5MjM5ADMwAjMwUzL3czL0UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![直積](/img/f/8ee/wZwpmLzUDOwEzMyAzN0YzM1UTM1QDN5MjM5ADMwAjMwUzLwczL1EzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![直積](/img/5/0a1/wZwpmL4QjN4MDO4czN0YzM1UTM1QDN5MjM5ADMwAjMwUzL3czL0UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
由定義可知, 是一個 塊的分塊矩陣,它是一個 行 列的矩陣, 與 有相同的階數,但一般 ≠ ,即矩陣的直積不滿足交換律。
由直積的定義容易推出以下定理。
(1) 兩個上三角陣的直積也是上三角陣;
(2) 兩個對角陣的直積仍是對角陣;
![直積](/img/d/324/wZwpmLyUDM4UzMxAzN0YzM1UTM1QDN5MjM5ADMwAjMwUzLwczLyIzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![直積](/img/7/1cc/wZwpmL4gTN3YzMxMTN0YzM1UTM1QDN5MjM5ADMwAjMwUzLzUzL1AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
(3) ( 為單位矩陣)。