漂亮的矩阵习题
- 格式:pdf
- 大小:90.75 KB
- 文档页数:3
一些漂亮的矩阵不等式西西提供(2011.09)
()()()
3213213213213212det 1
2det 12det 1)4(d 1)4(d 1)4(d 1)(,,1M M M M M M M M M M et M et M et Hermitian C M M M M n +++
+++++≥++∈:
矩阵且是正定的,证明是、设(
)(
)()()()()
(
)
2
3212
131132
32
3
2332222
2122121321det det det 222det 22det 22det )(,,2n n n n n n
n M M M M M M M M M M M M M M M M M M M M M Hermitian C M M M M ++≥+++++++++++∈矩阵且是正定的,证明
是、设()[]()
2
22333det 56det )(,,3C B A I C B A I C B A Hermitian C M C B A n n n n ++≥+++=++∈证明:矩阵且是正定的,满足是、设)
,)(,(912596789)(,422矩阵是非负正定则说如果矩阵是的矩阵,且是说明:如果证明:矩阵,且是、设Hermitian Y X Y X Hermitian C M Y X n n I I AB B A I B AB A Hermitian C M B A n n n n
n −≥∈×≤++=++∈0
)det()
)()(()
(,][,,,22004(0)(d ,][,,,54
4
3
3
2
2,1214
)
(4)(,1212
≥++++==≥++=
=≤≤+−
+−−≤≤B b b b b b b b b b b b b B n b b b A et a a e
e
a a A n a a a j i j i j i j i j i ij n j i ij n j
i a a a a a a ij n j i ij n j i j i j i 证明:且个正实数,且是):若(入学试题)年哈弗大学向国内招生证明:且个正实数,且是、设⋯⋯⋯⋯
()
()())
555det(14d :)2(,2det det 1,3)()()(,,6n 2222223
333C B A I ABC et ABC I CA BC AB I C B A AB BA A B AB Hetmitian C M C B A n n n ++≥++≤++=++−=−∈):(证明:
矩阵且是正定的,满足的、设n
AB Tr C Tr CA Tr B Tr BC Tr A Tr I C B A Hermitian C M C B A n n 6)()()()()()()(,,7444
≤+++++=++∈,证明:
矩阵且是正定的,满足是、设n
CA Tr A C Tr A C Tr BC Tr C B Tr C B Tr AB Tr B A Tr B A Tr I C B A Hermitian C M C B A n n 2012)
()()()
()()()
()()(1006)(,,82
2
222
2
222
2
22≤−+++
−+++
−++=++∈证明:
矩阵且是正定的满足是、设0
111111111det 11111111
1det 1,1921
2
12121
212122
2
11122
2
1112121>⎥
⎥⎥
⎥⎥⎦⎤
⎢⎢⎢⎢⎢⎣
⎡−−−−−−−−−+⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡+++++++++=<<<≤<<<≤M a a a a a a a a a a a a a a a a a a M b b b a a a n n n n n n b
n b
n b n b
b b b
b b b n b
n b n b b b b
b b n n 证明:且都是实数,记、设⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯()()[]()()[]()()[]()()[]
,)(,,,101
1
1
1
≥+−++−++−++−∈−−−−B A A D Tr A D D C Tr D C C B Tr C B B A Tr Hermitian C M D C B A n 证明:
矩阵且是正定的是、设
一些漂亮的矩阵恒等式
[]
[]
C
C B C I B B BA AB A B AB B A B AB b b b b b b a R M b B a A n n b b b k k n
j
i j i j i j i n n
j i ij
n
j i ij n ==++++++=++++=
∈==≥∞
→≤≤≤≤22223334
4
2
2,,1,121,lim 2226)(3,)()
(,)1(,,,1证明:令且有且有个正实数,且是、设⋯⋯0
)(6)(2)(10)(6,,)(,,2333222444666=++−++−++++++++∈CA BC AB A C C B B A C B A C B A C B A C B A Hermitian C M C B A n 使得
矩阵,求出所有是、设()()
是已知的正整数
其中计算:矩阵,满足、设n k m C B A P I C B A I C B A C B A
A B B A Hermitian C L C B A k k k n
m m m n n n n n n n n ,,,2011,,
)(G ,,3121212121212)12()12()12(12121
23553+++++++−+−+−+++++==++=++++=∈2012
201220122011
20112011,)(G ,,,,4cZ bY aX P XYZ ZX YZ XY cZ bY aX Hermitian C L Z Y X c b a n ++==++==∈计算矩阵,若是已知的实数,且、设()
2
|)det(|,422,0),(),(53
113=×++++=++=−−∈∈++B n n I I BA AB A B AB A A B AB I A A C M B R M A n n m m m m n n n 阶矩阵,求证:表示这里和且、设