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发信人: zhangghost (老七★野人部落★), 信区: kaoyan 标题: 相似矩阵发信站: 珞珈山水 (Sat Oct 2 00:36:14 2010), 站内 In linear algebra, two n-by-n matrices A and B are called similar if B=P^(-1)AP for some invertible n-by-n matrix P. Similar matrices represent the same lin ear transformation under two different bases, with P being the change of bas is matrix. The matrix P is sometimes called a similarity transformation. In the context of matrix groups, similarity is sometimes referred to as conjugacy, with si milar matrices being conjugate. [edit] Properties Similarity is an equivalence relation on the space of square matrices. Similar matrices share many properties: Rank Determinant Trace Eigenvalues (though the eigenvectors will in general be different) Characteristic polynomial Minimal polynomial (among the other similarity invariants in the Smith norma l form) Elementary divisors There are two reasons for these facts: Two similar matrices can be thought of as describing the same linear map, bu t with respect to different bases The map X ? P?1XP is an automorphism of the associative algebra of all n-by- n matrices, as the one-object case of the above category of all matrices. Because of this, for a given matrix A, one is interested in finding a simple "normal form" B which is similar to A—the study of A then reduces to the s tudy of the simpler matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex numbers (or any algebraically closed field), every m atrix is similar to a matrix in Jordan form. Another normal form, the ration al canonical form, works over any field. By looking at the Jordan forms or r ational canonical forms of A and B, one can immediately decide whether A and B are similar. The Smith normal form can be used to determine whether matri ces are similar, though unlike the Jordan and Frobenius forms, a matrix is n ot necessarily similar to its Smith normal form. [edit] Notes Similarity of matrices does not depend on the base field: if L is a field co ntaining K as a subfield, and A and B are two matrices over K, then A and B are similar as matrices over K if and only if they are similar as matrices o ver L. This is quite useful: one may safely enlarge the field K, for instanc e to get an algebraically closed field; Jordan forms can then be computed ov er the large field and can be used to determine whether the given matrices a re similar over the small field. This approach can be used, for example, to show that every matrix is similar to its transpose. In the definition of similarity, if the matrix P can be chosen to be a permu tation matrix then A and B are permutation-similar; if P can be chosen to be a unitary matrix then A and B are unitarily equivalent. The spectral theore m says that every normal matrix is unitarily equivalent to some diagonal mat rix. Specht's theorem states that two matrices are unitarily equivalent if a nd only if they satisfy certain trace equalities. 【在 huanglu () 的大作中提到: 】 : 相似： : 合同： : 等价： : ................... -- ※ 修改:·zhangghost 于 Oct 2 01:11:11 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: 相似矩阵
发信站: 珞珈山水 (Sat Oct  2 00:36:14 2010), 站内

In linear algebra, two n-by-n matrices A and B are called similar if

B=P^(-1)AP

for some invertible n-by-n matrix P. Similar matrices represent the same lin
ear transformation under two different bases, with P being the change of bas
is matrix.

The matrix P is sometimes called a similarity transformation. In the context
of matrix groups, similarity is sometimes referred to as conjugacy, with si
milar matrices being conjugate.

[edit] Properties

Similarity is an equivalence relation on the space of square matrices.

Similar matrices share many properties:

Rank

Determinant

Trace

Eigenvalues (though the eigenvectors will in general be different)

Characteristic polynomial

Minimal polynomial (among the other similarity invariants in the Smith norma
l form)

Elementary divisors

There are two reasons for these facts:

Two similar matrices can be thought of as describing the same linear map, bu
t with respect to different bases

The map X ? P?1XP is an automorphism of the associative algebra of all n-by-
n matrices, as the one-object case of the above category of all matrices.

Because of this, for a given matrix A, one is interested in finding a simple
"normal form" B which is similar to A—the study of A then reduces to the s
tudy of the simpler matrix B. For example, A is called diagonalizable if it
is similar to a diagonal matrix. Not all matrices are diagonalizable, but at
least over the complex numbers (or any algebraically closed field), every m
atrix is similar to a matrix in Jordan form. Another normal form, the ration
al canonical form, works over any field. By looking at the Jordan forms or r
ational canonical forms of A and B, one can immediately decide whether A and
B are similar. The Smith normal form can be used to determine whether matri
ces are similar, though unlike the Jordan and Frobenius forms, a matrix is n
ot necessarily similar to its Smith normal form.

[edit] Notes

Similarity of matrices does not depend on the base field: if L is a field co
ntaining K as a subfield, and A and B are two matrices over K, then A and B
are similar as matrices over K if and only if they are similar as matrices o
ver L. This is quite useful: one may safely enlarge the field K, for instanc
e to get an algebraically closed field; Jordan forms can then be computed ov
er the large field and can be used to determine whether the given matrices a
re similar over the small field. This approach can be used, for example, to
show that every matrix is similar to its transpose.

In the definition of similarity, if the matrix P can be chosen to be a permu
tation matrix then A and B are permutation-similar; if P can be chosen to be
a unitary matrix then A and B are unitarily equivalent. The spectral theore
m says that every normal matrix is unitarily equivalent to some diagonal mat
rix. Specht's theorem states that two matrices are unitarily equivalent if a
nd only if they satisfy certain trace equalities.

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

--

※ 修改:·zhangghost 于 Oct  2 01:11:11 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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