设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]列满秩矩阵，它可分解为[tex=3.143x1.214]88XYSTidWfFV3HXEL1LUxQ==[/tex]，其中[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]是列向量组为正交单位向量组的[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]矩阵，[tex=0.786x1.0]as0RCzgUx1oS48cKHRAVVg==[/tex]为主对角都为正数的上三角矩阵.证明对于任意[tex=6.5x1.429]U9x/V18E+mu1qwYsBmBQPcvmo3Olsakqdy19fdc0TLWnUN4vVKSoEwzw0JukSV9NIMYCCwBaBdG+Ew1xF9aWYbzVEM+LdJE6Xb69L7wQmps=[/tex]是线性方程组[tex=4.929x1.357]Nivl9w+4kNkx9bsbdtPk3fYmCCXUkRkNNgH31iK9Uwg=[/tex]的唯一解.

2022-06-29

设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是实数域上的[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]列满秩矩阵，它可分解为[tex=3.143x1.214]88XYSTidWfFV3HXEL1LUxQ==[/tex]，其中[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]是列向量组为正交单位向量组的[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]矩阵，[tex=0.786x1.0]as0RCzgUx1oS48cKHRAVVg==[/tex]为主对角都为正数的上三角矩阵.证明对于任意[tex=6.5x1.429]U9x/V18E+mu1qwYsBmBQPcvmo3Olsakqdy19fdc0TLWnUN4vVKSoEwzw0JukSV9NIMYCCwBaBdG+Ew1xF9aWYbzVEM+LdJE6Xb69L7wQmps=[/tex]是线性方程组[tex=4.929x1.357]Nivl9w+4kNkx9bsbdtPk3fYmCCXUkRkNNgH31iK9Uwg=[/tex]的唯一解.

答案：

证明：设[tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex]的列向量为[tex=5.429x1.071]vWBewMbNQkISGdBsNA+6z0gEDH5lNCzAIthFKYpM2FK25wTO3yS2cGzzDDNuHUrr9xFodQiN5Ef1ph3LZV/vl/ovTrGS0LpH0As0uPos7Nk=[/tex].则.[tex=12.643x5.929]y8FyjFcHvT3XMbY3QII+HqwgWIic9gD84qGLdD4mHl9pbEzAqjMN7WbCaStVdwlHbxZdqBp0Nuwur2NtHUT9bg8MkIKv0yNUIKmHwttZnLuO7bTJFMy8Zut/l1cldQYryHpOCK8T+liX05zjaXzLAqXrhCeGXJwiUGQTJb4qvFldW6rCTBSPzvBlBdDp0E0CkmZ0dNP1QsYUU0DKTMAg8n41DNdM4FrTFaQ6Eyikim01KiZpxY6zNOXKUS3JSYKKP2D2MG1fLslQgy+if73PYW/SHi9kqxmFJ9SOb6ruSzGdka0ATuzS8lgBK0a09hNVL3UfYFJ7yfFXK52kHd5cCQ==[/tex][tex=14.786x5.786]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[/tex].从而[tex=24.143x2.0]v/M/cNj2O45fbaYT8C7+4ZyIq8Fj59tUorlmwqGQ1xSf56dSNcnO0+W7X0B+q8+5l6I21SaetbKs1JvaoGB9zcDqDUZ08fUx1zlrizglicuOrRDqum0QHD41e9Lb1tCQ5jjxI1GjpDNomC9GXXdXKM8koFeE8YKhgKvhdrxI5gdUKVyseQPqE4aAsLaC2l7BGVWVipXPEonWgguLacrz9eVrXTk4pn+GbKcSJceQ958uiDFxAH7bNHJa4pu2FgMYDw38cBddsOnU07SnJZaaiQ==[/tex][tex=9.643x1.929]Vh3EpgYamXw37N4sM6em0P24etGk3pSE0ostTez+gCiakvfZuVKD2yfVOzMPALpdbxly4htMjZ/tqk1bI3O+MDDf7B+mgahQe/kO+QHHaPEBHsXoSla1GMyDU1TybOo2wcwE+YYUAoaKss+HF89VEQ==[/tex].因此[tex=3.286x1.429]nyetI73TkIAkVZ7bpeEtVquELdDXe0Vtegujlqxdi2s=[/tex]是线性方程组[tex=4.929x1.357]Nivl9w+4kNkx9bsbdtPk3fYmCCXUkRkNNgH31iK9Uwg=[/tex]的一个解.由于[tex=10.643x1.429]k8PvTJe4iQVkvPfhUhxDGCk2kSGdYRGbGNYqRQoKDREIedYK5xb+mKt0m2SB9lyXfdxF04Fr1ivWRfJsZJP5x5SnrnW0V8m8GiJx8DajU+Y=[/tex]，因此[tex=4.143x1.429]/HMWc3f2ge4oAJ8LZZxcLo49Kpgseyi3VU7lFPlsgkqaguHZcArDFflHLgv8MIXv[/tex].从而线性方程组[tex=4.929x1.357]Nivl9w+4kNkx9bsbdtPk3fYmCCXUkRkNNgH31iK9Uwg=[/tex]有唯一解，它就是[tex=3.286x1.429]nyetI73TkIAkVZ7bpeEtVquELdDXe0Vtegujlqxdi2s=[/tex].

举一反三

内容

0
试写出下述二次规划的[tex=2.214x1.143]pxkUhTl2yMbkNlgeyYXq6A==[/tex]条件：[tex=11.143x4.643]a0s3MH7cLIdmiBRR0YN06+tn4IA9xf+I0MkjOgk1z3GoG877R4tY7Zi6FsKt7hPd08rvAiE+M6LvOeRwN0i0ebm2uxEI2JOW9lB7QV42iI9zR4/F3ELA1PFKDPMHiKspJAYJf1WEdPcfPRc+wYVvaCelGjurP63M4Q9PYBkLXoGwhr2uxqF0fHe2iO8emw5zZRCejJ+2lmeRf67MG4XCArjxZOmSveey3HhPDKQrGhg=[/tex]其中[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]为[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]矩阵，[tex=0.857x1.0]h610M+sGyf59WggKwaDo1Q==[/tex]为[tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex]矩阵，[tex=0.714x1.0]J/aA9EEo0KmJFnWWfX7LmQ==[/tex]为[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维列向量，[tex=0.429x1.0]JThLUuJ8WswSAPiYZWihWg==[/tex]为[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex]维列向量，变量[tex=0.857x1.0]N7iCrOsS+NNEUUlnsYCi1g==[/tex]为[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]维列向量。
1
证明 :设[tex=2.786x1.143]sJiVcoTfEg/JbhJV/202TA==[/tex]矩阵 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的秩为[tex=0.786x1.0]3aIfIj/PvpRDhDMMRyp3Yw==[/tex] 则有[tex=2.571x1.071]v4dmMOo3Ht85R401A97p+g==[/tex] 的列满秩矩阵[tex=0.786x1.0]eh2CuRqBLsAEAbb2XRxaBg==[/tex] 和[tex=2.286x1.071]5AfSV6NTVwiHny+StJ+UCA==[/tex]的行满秩矩阵 [tex=0.857x1.214]ChdusW5rAupjge6v/DGHRA==[/tex] 使 [tex=3.286x1.214]Jxd8pQJL4d8RyMjmHZyNcQ==[/tex]
2
证明：如果[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]级正交矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是上三角矩阵，那么[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是对角矩阵，且[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的主对角元为1或-1.
3
设A, B都是[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]矩阵,证明A~B的充分必要条件是[tex=5.643x1.357]SckQ0Wk1HqsZFefj450OCg==[/tex]
4
设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]为[tex=0.929x0.786]D9maNLyVVGrC3QbL9jjRWg==[/tex]阶实对称阵．且正定，[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]为[tex=2.714x1.071]319e/AVA5VexfWBQXpJ9ug==[/tex]实矩阵，[tex=1.071x1.429]FW17NNOy+nNs0P4RJiU76Q==[/tex]为[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的转置矩阵，试证[tex=2.571x1.429]uCtEJJAWUMGXtgmW8PsAZg==[/tex]为正定矩阵的充分必要条件是秩[tex=3.214x1.357]9Wef3TrL8ArBiBvvnB4k/g==[/tex]