定義
![矩陣相似](/img/6/eb7/wZwpmLxIzMxEDM0gTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL3YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
設A,B為數域P上兩個n階矩陣,如果可以找到數域P上的n階可逆矩陣X,使得 ,則稱A相似於B,記作A~B。
性質
(1) 若A相似於B,則A等價於B(即A可通過初等變換化為B)
(2) 若A相似於B,則tr(A)=tr(B)
(3) 若A相似於B,則|A|=|B|
以上三條反之皆不成立
![矩陣相似](/img/0/b9f/wZwpmL4cjM1IjNwADN3QTN1UTM1QDN5MjM5ADMwAjMwUzLwQzLyUzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
證明:若A~B,則存在可逆矩陣X使得
![矩陣相似](/img/1/fd9/wZwpmL2ITN5UDOxYDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzLzQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![矩陣相似](/img/6/d4d/wZwpmLxcDM1EjM1gjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4IzL0UzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![矩陣相似](/img/a/47c/wZwpmL3MzNwEDN0UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzL0MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
令 , 則 ,即A與B等價,(1)成立
![矩陣相似](/img/6/eb7/wZwpmLxIzMxEDM0gTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL3YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
因 ,於是有
![矩陣相似](/img/5/03f/wZwpmL4UjN3MzM3kjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5IzL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
即A與B有相同特徵值多項式,因而有相同的特徵值,故(2)(3)也成立。
![矩陣相似](/img/b/ff8/wZwpmL2QTOyQzM1YjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2IzLwEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![矩陣相似](/img/a/974/wZwpmLxMzNxATOzcjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzLzYzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
反例: ,
![矩陣相似](/img/a/7d4/wZwpmLyADMwATM2EzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxMzLxgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
顯然A與B等價,並且tr(A)=tr(B),|A|=|B|,但A與B不可能相似(因A=E,對任意的n階可逆矩陣X,都有 )。
相似是矩陣間的一種重要關係,這種關係具有以下三個性質:
![矩陣相似](/img/3/d91/wZwpmL3cDMxEjM3gTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzLwIzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/3/18e/wZwpmL3MTNzEDN2kTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5EzL3EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![矩陣相似](/img/b/b17/wZwpmLyIDN2UTN1MzNwIDN0UTMyITNykTO0EDMwAjMwUzLzczL1MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/8/6bc/wZwpmL3ATOwYTM3gDN0MTN1UTM1QDN5MjM5ADMwAjMwUzL4QzLwYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/5/e9e/wZwpmLxADO3kzMzkDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzL2AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![矩陣相似](/img/8/6bc/wZwpmL3ATOwYTM3gDN0MTN1UTM1QDN5MjM5ADMwAjMwUzL4QzLwYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/6/eb7/wZwpmLxIzMxEDM0gTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL3YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/c/8fd/wZwpmLxgTM2UjMxczM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3MzLyAzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/8/7ee/wZwpmL0cTO5cjMyEjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLxIzL3gzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![矩陣相似](/img/5/e9e/wZwpmLxADO3kzMzkDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5AzL2AzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![矩陣相似](/img/8/6bc/wZwpmL3ATOwYTM3gDN0MTN1UTM1QDN5MjM5ADMwAjMwUzL4QzLwYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/c/ac2/wZwpmLyYTMygzM1IzN0MTN1UTM1QDN5MjM5ADMwAjMwUzLyczLyUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/8/22d/wZwpmLwYDO1kTNxAzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLwMzLwUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/8/6bc/wZwpmL3ATOwYTM3gDN0MTN1UTM1QDN5MjM5ADMwAjMwUzL4QzLwYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/c/ac2/wZwpmLyYTMygzM1IzN0MTN1UTM1QDN5MjM5ADMwAjMwUzLyczLyUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/6/eb7/wZwpmLxIzMxEDM0gTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4EzL3YzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/2/0a6/wZwpmLwYDM5cTM4YzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzLzczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![矩陣相似](/img/8/0cd/wZwpmLyITOwUzN4UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzLxQzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![矩陣相似](/img/7/557/wZwpmL1ETNzYTOzEDN3QTN1UTM1QDN5MjM5ADMwAjMwUzLxQzLwgzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![矩陣相似](/img/8/22d/wZwpmLwYDO1kTNxAzM3QTN1UTM1QDN5MjM5ADMwAjMwUzLwMzLwUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
1.反身性: 。這是因為 (其中 為單位矩陣,下同)。2.對稱性:如果 ,那么 。事實上如果 ,那么有X使 ,令 ,就有 ,所以 。3.傳遞性:如果 , ,那么 。因為若,,即存在X,Y使 , 。令 ,就有 ,因此有 。(具有以上三個性質的關係統稱為等價關係)
矩陣相似充分必要條件
![矩陣相似](/img/2/d94/wZwpmL2czM0QzMyUjNxIDN0UTMyITNykTO0EDMwAjMwUzL1YzLyYzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
設A,B是數域P上兩個 矩陣:
![矩陣相似](/img/f/679/wZwpmLyMjMwMTO1kzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL5MzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![矩陣相似](/img/b/511/wZwpmLxQDM0kDM1IjM3QTN1UTM1QDN5MjM5ADMwAjMwUzLyIzL1EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(1) A與B相似的充分必要條件是它們的特徵矩陣 與 等價。
(2) A與B相似的充分必要條件是它們有相同的不變因子。
(3) 兩個同級複數矩陣相似的充分必要條件是它們有相同的初等因子。
套用
![矩陣相似](/img/a/7c8/wZwpmLyITMzAzNxgDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL4AzL0EzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![矩陣相似](/img/4/7e3/wZwpmL1IDO5EDN4YDM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2AzL0MzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
例:已知A~B,其中 , 。
![矩陣相似](/img/1/2cb/wZwpmLwAjN4ITM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![矩陣相似](/img/5/9b3/wZwpmL3ATM1MzN4UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzLxQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
(1)求a,b的值;(2)求可逆矩陣X使 ;(3)求 。
解:(1)因tr(A)=tr(B)及|A|=|B|可得a=5,b=3
(2)顯然A的特徵值為2,1,5,即B的特徵值也為2,1,5
![矩陣相似](/img/5/c5a/wZwpmL4QDOycjNzYzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL2MzLxEzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![矩陣相似](/img/4/2c8/wZwpmLwgTOzEzMyQTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL0EzL0EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![矩陣相似](/img/b/b64/wZwpmLxMzN2cDO2cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL4QzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![矩陣相似](/img/2/bff/wZwpmLwADN4EjM3cjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3IzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
由 可得對應特徵向量 , ,
![矩陣相似](/img/c/327/wZwpmLwcDM4kjN2UjM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1IzLzQzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![矩陣相似](/img/1/2cb/wZwpmLwAjN4ITM3cTM3QTN1UTM1QDN5MjM5ADMwAjMwUzL3EzL2MzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
令 ,則有 。
![矩陣相似](/img/9/ab3/wZwpmLwUTM1MTN1UzM3QTN1UTM1QDN5MjM5ADMwAjMwUzL1MzLyEzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
(3) 。