证明：对任意n阶矩阵[tex=4.857x1.429]MmeklMP/j6saWqycN/i3Z3wqMHuQmIjSWcnVpwDa/28sKs+vgCQyhxOwRCKhyh7sjdwGTeaCYmH0lW9izQdVxg==[/tex]为对称据矩阵，[tex=3.071x1.357]EOQ1Rjmvh2eNjKR72INhKGvD/DKi/KRYjyVJEbwB31BEI8bbGlHTf/dugNu8h/fs[/tex]为反对称矩阵.
证明：对任意n阶矩阵[tex=4.857x1.429]MmeklMP/j6saWqycN/i3Z3wqMHuQmIjSWcnVpwDa/28sKs+vgCQyhxOwRCKhyh7sjdwGTeaCYmH0lW9izQdVxg==[/tex]为对称据矩阵，[tex=3.071x1.357]EOQ1Rjmvh2eNjKR72INhKGvD/DKi/KRYjyVJEbwB31BEI8bbGlHTf/dugNu8h/fs[/tex]为反对称矩阵.
证明: 设 [tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 都是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶正交方阵,则[tex=2.714x1.357]RRZ9zlAN4pWdGS7d9wHOksQIkUN+jWMfk9arM96kBXA=[/tex] 或 [tex=1.286x1.143]Mj6+lbt3rBoas+xQLVX/oA==[/tex]
证明: 设 [tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 都是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶正交方阵,则[tex=2.714x1.357]RRZ9zlAN4pWdGS7d9wHOksQIkUN+jWMfk9arM96kBXA=[/tex] 或 [tex=1.286x1.143]Mj6+lbt3rBoas+xQLVX/oA==[/tex]
证明: 设[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex]都是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶正交方阵,则 [tex=5.571x1.429]317mMb/UfJBjZHDU7raSnlGWNv3TAfOvDYKp6rxGdYH/wkTdLKG3lnIOSFYz8youND3JkA/f56Zt6vj//KbUBaNtSDrFJ/TojTxEfphc2zw=[/tex] 也是正交方阵
证明: 设[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex]都是 [tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶正交方阵,则 [tex=5.571x1.429]317mMb/UfJBjZHDU7raSnlGWNv3TAfOvDYKp6rxGdYH/wkTdLKG3lnIOSFYz8youND3JkA/f56Zt6vj//KbUBaNtSDrFJ/TojTxEfphc2zw=[/tex] 也是正交方阵
设 [tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 均为[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶方阵,证明 [tex=1.786x1.0]96v/QovsZC+SUffULwqQnvkECUDsFAQTkNLNhDaRrpw=[/tex] 与 [tex=1.786x1.0]kvRlRS1CLNU1fC2siJ4VO2nX6gBe0vc1kFMLW9VYZjM=[/tex] 有相同的特征值.
设 [tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 均为[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex] 阶方阵,证明 [tex=1.786x1.0]96v/QovsZC+SUffULwqQnvkECUDsFAQTkNLNhDaRrpw=[/tex] 与 [tex=1.786x1.0]kvRlRS1CLNU1fC2siJ4VO2nX6gBe0vc1kFMLW9VYZjM=[/tex] 有相同的特征值.
[tex=1.857x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 皆为 [tex=2.429x1.071]/1altEfb7YJTb68/W6737A==[/tex] 复矩阵,证明:方程 [tex=4.0x1.0]iBuDKI65/og8QoaCBkLEgw==[/tex] 有非零解的 充分必要条件是[tex=2.0x1.214]Fu8IbDhz8zi1Fz+2a1sLzg==[/tex]有公共特征值.
[tex=1.857x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 皆为 [tex=2.429x1.071]/1altEfb7YJTb68/W6737A==[/tex] 复矩阵,证明:方程 [tex=4.0x1.0]iBuDKI65/og8QoaCBkLEgw==[/tex] 有非零解的 充分必要条件是[tex=2.0x1.214]Fu8IbDhz8zi1Fz+2a1sLzg==[/tex]有公共特征值.
设[tex=3.143x1.214]MmeklMP/j6saWqycN/i3Z/a69BxU/XzAAmVLUDwLAKYsGROs6hwqumtrrypVwJLZ[/tex]是 [tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex] 方阵[tex=5.786x1.214]uSJxMvQZ0VWTKkbNd1sz3Q==[/tex]试证如果[tex=9.857x1.357]KNz1FY9OAxkHJ6m0crUOlLQnAzLePqaEkKGo26MdANc=[/tex]则[tex=12.5x1.357]j1UW3kvoxPwU8IkDv6BF1SHztaQn6enIK4q8bKGDloHemP5HRVc3eNHhd4woEIvDIvuc6phH6qy9P2F6L/fvpQLm96yU+JQuEYsY3bG3E3dHzUBojYCTzotgWbwyzLLPj4wbQCYUoB7GMqhgm7Hli1Zpc08vZa3rB/3FFf4H+NnLwlFAA5mB2/6FIASxpxyh[/tex]并计算 [tex=4.286x1.143]Rt43yfj/R7vZbskTbnkKRYw4Bhs7CQqPUDVH0KrKY0rpESjCsltPX9XuAUmcYzd4[/tex]
设[tex=3.143x1.214]MmeklMP/j6saWqycN/i3Z/a69BxU/XzAAmVLUDwLAKYsGROs6hwqumtrrypVwJLZ[/tex]是 [tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex] 方阵[tex=5.786x1.214]uSJxMvQZ0VWTKkbNd1sz3Q==[/tex]试证如果[tex=9.857x1.357]KNz1FY9OAxkHJ6m0crUOlLQnAzLePqaEkKGo26MdANc=[/tex]则[tex=12.5x1.357]j1UW3kvoxPwU8IkDv6BF1SHztaQn6enIK4q8bKGDloHemP5HRVc3eNHhd4woEIvDIvuc6phH6qy9P2F6L/fvpQLm96yU+JQuEYsY3bG3E3dHzUBojYCTzotgWbwyzLLPj4wbQCYUoB7GMqhgm7Hli1Zpc08vZa3rB/3FFf4H+NnLwlFAA5mB2/6FIASxpxyh[/tex]并计算 [tex=4.286x1.143]Rt43yfj/R7vZbskTbnkKRYw4Bhs7CQqPUDVH0KrKY0rpESjCsltPX9XuAUmcYzd4[/tex]
设 [tex=7.143x2.786]g+/KVfaQxdXa8hxnv147pHWU52BSD9LdtB9aAf9v+Sbw0blXLyIcHBU0jSX5ERGgSQrByRePHgOQ3Dstda1AfQKRafS7BreZoZMXKC8Vbf2UFrXzBeXPcDyK1mfGT//NllwrWZYMcqn7M6gzUNt+dxT4qlQcWezHyfAwkusJil4=[/tex] 其中 [tex=2.0x1.214]p/fPb4cKwKYaAJ8NhtZPtw==[/tex]分别是 [tex=1.929x1.0]FLsL1n4WDTNpV4dn6Kq2dg==[/tex] 阶矩阵 求证:若[tex=0.929x1.0]dS4Ce7aCn5Z2jKx4QASmCg==[/tex] 是正定矩阵[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 都是正定矩阵,反之也成立.
设 [tex=7.143x2.786]g+/KVfaQxdXa8hxnv147pHWU52BSD9LdtB9aAf9v+Sbw0blXLyIcHBU0jSX5ERGgSQrByRePHgOQ3Dstda1AfQKRafS7BreZoZMXKC8Vbf2UFrXzBeXPcDyK1mfGT//NllwrWZYMcqn7M6gzUNt+dxT4qlQcWezHyfAwkusJil4=[/tex] 其中 [tex=2.0x1.214]p/fPb4cKwKYaAJ8NhtZPtw==[/tex]分别是 [tex=1.929x1.0]FLsL1n4WDTNpV4dn6Kq2dg==[/tex] 阶矩阵 求证:若[tex=0.929x1.0]dS4Ce7aCn5Z2jKx4QASmCg==[/tex] 是正定矩阵[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 都是正定矩阵,反之也成立.
设[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 是两个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶实对称矩阵证明 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 相似的充要条件是 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 有相同的特 征值.
设[tex=2.071x1.214]MmeklMP/j6saWqycN/i3Z6f4PxINUzvl/cZhR6Tpp74=[/tex] 是两个[tex=0.643x0.786]SBMIs+VUk7//BOpfqlQl0w==[/tex]阶实对称矩阵证明 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 相似的充要条件是 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 有相同的特 征值.