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发信人: zhangghost (老七★野人部落★), 信区: kaoyan 标题: 等价矩阵发信站: 珞珈山水 (Sat Oct 2 01:01:45 2010), 站内 In linear algebra, two rectangular m-by-n matrices A and B are called equiva lent if B=Q^(-1)AP for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Sim ilar matrices represent the same linear transformation V → W under two diff erent choices of a pair of bases of V and W, with P and Q being the change o f basis matrices in V and W respectively. The notion of equivalence should not be confused with that of similarity, wh ich is only defined for square matrices, and is much more restrictive (simil ar matrices are certainly equivalent, but equivalent square matrices need no t be similar). That notion corresponds to matrices representing the same end omorphism V → V under two different choices of a single basis of V, used bo th for initial vectors and their images. [edit] Properties Matrix equivalence is an equivalence relation on the space of rectangular ma trices. For two rectangular matrices of the same size, their equivalence can also be characterized by the following conditions The matrices can be transformed into one another by a combination of element ary row and column operations. The matrices have the same rank. 【在 huanglu () 的大作中提到: 】 : 相似： : 合同： : 等价： : ................... -- ※ 修改:·zhangghost 于 Oct 2 01:11:28 2010 修改本文·[FROM: 125.220.141.] ※ 来源:·珞珈山水 bbs.whu.edu.cn·[FROM: 125.220.141.]
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发信人: zhangghost (老七★野人部落★), 信区: kaoyan
标  题: 等价矩阵
发信站: 珞珈山水 (Sat Oct  2 01:01:45 2010), 站内

In linear algebra, two rectangular m-by-n matrices A and B are called equiva
lent if

B=Q^(-1)AP

for some invertible n-by-n matrix P and some invertible m-by-m matrix Q. Sim
ilar matrices represent the same linear transformation V → W under two diff
erent choices of a pair of bases of V and W, with P and Q being the change o
f basis matrices in V and W respectively.

The notion of equivalence should not be confused with that of similarity, wh
ich is only defined for square matrices, and is much more restrictive (simil
ar matrices are certainly equivalent, but equivalent square matrices need no
t be similar). That notion corresponds to matrices representing the same end
omorphism V → V under two different choices of a single basis of V, used bo
th for initial vectors and their images.

[edit] Properties

Matrix equivalence is an equivalence relation on the space of rectangular ma
trices.

For two rectangular matrices of the same size, their equivalence can also be
characterized by the following conditions

The matrices can be transformed into one another by a combination of element
ary row and column operations.

The matrices have the same rank.

【在 huanglu () 的大作中提到: 】
: 相似：
: 合同：
: 等价：
: ...................

--

※ 修改:·zhangghost 于 Oct  2 01:11:28 2010 修改本文·[FROM: 125.220.141.*]
※ 来源:·珞珈山水 bbs.whu.edu.cn·[FROM: 125.220.141.*]

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