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发信人: zhangghost (老七★野人部落★), 信区: kaoyan 标题: 合同矩阵发信站: 珞珈山水 (Sat Oct 2 00:54:10 2010), 站内 In mathematics, two matrices A and B over a field are called congruent if th ere exists an invertible matrix P over the same field such that PTAP = B where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on t he Gram matrix attached to a bilinear form or quadratic form on a finite-dim ensional vector space: two matrices are congruent if and only if they repres ent the same bilinear form with respect to different bases. Note that Halmos[1] defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this d efinition has not been adopted by most other authors. Congruence over the reals Sylvester's law of inertia states that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenval ues. That is, the number of eigenvalues of each sign is an invariant of the associated quadratic form.[2] 【在 huanglu () 的大作中提到: 】 : 相似： : 合同： : 等价： : ................... -- ※ 修改:·zhangghost 于 Oct 2 01:11:21 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: 合同矩阵
发信站: 珞珈山水 (Sat Oct  2 00:54:10 2010), 站内

In mathematics, two matrices A and B over a field are called congruent if th
ere exists an invertible matrix P over the same field such that

PTAP = B

where "T" denotes the matrix transpose. Matrix congruence is an equivalence
relation.

Matrix congruence arises when considering the effect of change of basis on t
he Gram matrix attached to a bilinear form or quadratic form on a finite-dim
ensional vector space: two matrices are congruent if and only if they repres
ent the same bilinear form with respect to different bases.

Note that Halmos[1] defines congruence in terms of conjugate transpose (with
respect to a complex inner product space) rather than transpose, but this d
efinition has not been adopted by most other authors.

Congruence over the reals

Sylvester's law of inertia states that two congruent symmetric matrices with
real entries have the same numbers of positive, negative, and zero eigenval
ues. That is, the number of eigenvalues of each sign is an invariant of the
associated quadratic form.[2]

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

--

※ 修改:·zhangghost 于 Oct  2 01:11:21 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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