证明：对任一[tex=2.429x1.071]jLyhB8GAUqIuDKvKM/p5zw==[/tex]复系数矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] ，存在可逆矩阵[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]，使[tex=3.0x1.214]rN84CqmtCk5MRAP5g+8tJQ==[/tex]是上三角矩阵．

2022-06-30

证明：对任一[tex=2.429x1.071]jLyhB8GAUqIuDKvKM/p5zw==[/tex]复系数矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex] ，存在可逆矩阵[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]，使[tex=3.0x1.214]rN84CqmtCk5MRAP5g+8tJQ==[/tex]是上三角矩阵．

答案：

证: 由文献［１］Ｐ.313 定理 13 知，存在一组基[tex=12.5x1.071]uh36dzsJfggBlWeE2D47K+qxw+o4AxAaQnKZV+xY8vsLrQzfiATFyfLV9+mDwpI+PB7zwa0rXkckkYf9ohzCdkgxT8zFk3HF4Kt5sPB98tlgD7XUZSsy+s7h+/NBXSwH7jyPJl0jdmSVm+lEEXWzdjlqXF/bwC88DvPwsPBThrGtwxIQCekY1v6csdP615Jz[/tex]，使与矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]相应的线性变换[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的矩阵成若尔当标准形，从而[tex=8.286x4.214]fnpmC2J6JmQBLyo5NmGAzzRJB/jqXUzC0ef5Vpr+uhnC49wOJ0tSQwZpI8GgGXushEkpXKmuHJbxu4CO+qCZJVfFL1xe1LCjGkMqfrlCAEBycz6PsCJrqcfYUfsHN9tc9WYGzJAjBfzXu6YUcGU2YVQ3LfgfTSqqe/3w1SPx5wLMuiZrSUhfDOQ+JX3y7q0T7rsbTxm69E9aWe3V71d4FGC1ivSaOKwVs+secUOM2bFFYMh35fFsRDwpKxbts85m[/tex]    ......[tex=8.214x4.214]fnpmC2J6JmQBLyo5NmGAzzRJB/jqXUzC0ef5Vpr+uhnm2Ek2d2XHQs3Dx7Ox3K6H2LGSILLxWCgeZx9L3MYYbugjXiA9LByaGp7a6ukByC9d08KVh+XJTvu/eWpM2YN5xORL1ezlQQ07wqA/1XT9iqMW8V28zoRAMYV3tpYKOwWPxT820XzBcEN1tXHhS4nBCOQj8RAZL0Zz67wB62kLP3U3sRoHSNP6K0N67mY7kKK9WfyRCnKU26XwGK1AJ2C/[/tex]若过渡矩阵为[tex=0.643x1.0]Ft8KOBgb78fnKY0jEt4Rsg==[/tex]，则[tex=3.929x1.214]faiSE3MwKhL7XDJpoiAJPQ==[/tex][tex=2.357x1.286]S9jzQ3LTF4sucOPWlXPFFw==[/tex][tex=20.071x10.214]ThIgXOUyLYK/6IHx89IH/v1uNE/vivYL3C0goS322oDZ7NT6kSGQ/mun894Z8UXcPHPlOFv0uWpOZJtxeRreCyUDn6G8HjTYqtc26Nuuwk9ifkd8JIg4O+veEerXHSCZ4Ue+p1i+Mk+FPYP4JemZumcouBP29vOqCZKmiFJbdOY2oJVYNdlAFQf1hK+cBkWS10eU8BG1xi9HOkSc27PO14MsNnyoXUl9osRc4YfnGS2AzipRThwz5AyNXlXpC3EWKRQkC6UebZKpD1R3f3eyVevh6xAroDh2PVgRja6qWsIPaU1GWLEeZzC38Un99vLE0VW+GWq9wZ/SbBB0UxxIgppRdip1/w4AIwhmg8RDe18uVAKDCs5Vok6LGXHU8T+LWedOLD3oub9qM8f/zVNkCd9OkaTWaRwl6MInlSo4rPZJ1H14iFufIvS6PZ7Lku5wkgBF2h5gim77Nmg9YTL9lOKy/xrVZtNWHlhVRELgfPM=[/tex]今考虑新基[tex=12.857x1.071]uh36dzsJfggBlWeE2D47K+SEaqDMPs+95Xus6jaYS6bJDXVJ0E3g2NLo+tO4WGQBriZ3gKKHQfzTF/O+DcL2qbG4JWuXPF372OAYoKoWDzGRs3Otis6n8HV0pzJu4YPazD45lByEkff/IO2CTAPC5QRBa2bEJOHTSde8koMVzYFq4wbjlMf+Q6OK7dER7eiSihfnqGroO4ElxYUAJO8a6A==[/tex]，过渡矩阵为[tex=12.357x5.786]5ZuEj1KRR/p9rD5ciF5Q2qrrc5kuUs60+BS5rIgRKnpbuis3ICndMLPiW2fCx2PLd61QGaaBfU8HmPpSK4eeOiMnbDbpA5g/gDASOTDKumejIWRTfvx/n51TBKi69prNLbyArzCZgAiUaqRmjMbGg2X0cM1w9WJyNiYHDVbmITZx7tgmHbDWku3diECsFXoB[/tex]，其中[tex=17.071x4.929]zGVLMizKCKlYO4QKFkZBRxNZfQvgh4l/2LkGZ5VHhvx8bj0kZLANoMZAdDinkPKVV1NEHNxHKCDZ5ZwG0euNSzuOgWAQjIDtW1Xu1P7+jQjjZKSzhhKdaYfLcAzL+uoocWiEqG3YNjnm4RhoUx48X+1SNze5C7t26Q2nvp8uNQ3SjyeuwSrSKjKttnSNPZ2/[/tex]由[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]([tex=12.857x1.071]uh36dzsJfggBlWeE2D47K+SEaqDMPs+95Xus6jaYS6bJDXVJ0E3g2NLo+tO4WGQBriZ3gKKHQfzTF/O+DcL2qbG4JWuXPF372OAYoKoWDzGRs3Otis6n8HV0pzJu4YPazD45lByEkff/IO2CTAPC5QRBa2bEJOHTSde8koMVzYFq4wbjlMf+Q6OK7dER7eiSihfnqGroO4ElxYUAJO8a6A==[/tex])[tex=0.786x0.643]NhuTNiqjImitwKaHFutGOg==[/tex]([tex=12.857x1.071]uh36dzsJfggBlWeE2D47K+SEaqDMPs+95Xus6jaYS6bJDXVJ0E3g2NLo+tO4WGQBriZ3gKKHQfzTF/O+DcL2qbG4JWuXPF372OAYoKoWDzGRs3Otis6n8HV0pzJu4YPazD45lByEkff/IO2CTAPC5QRBa2bEJOHTSde8koMVzYFq4wbjlMf+Q6OK7dER7eiSihfnqGroO4ElxYUAJO8a6A==[/tex])[tex=20.071x10.214]ThIgXOUyLYK/6IHx89IH/v1uNE/vivYL3C0goS322oDZ7NT6kSGQ/mun894Z8UXcPHPlOFv0uWpOZJtxeRreCyUDn6G8HjTYqtc26Nuuwk9ifkd8JIg4O+veEerXHSCZ4Ue+p1i+Mk+FPYP4JemZumcouBP29vOqCZKmiFJbdOY2oJVYNdlAFQf1hK+cBkWS10eU8BG1xi9HOkSc27PO14MsNnyoXUl9osRc4YfnGS2AzipRThwz5AyNXlXpC3EWKRQkC6UebZKpD1R3f3eyVevh6xAroDh2PVgRja6qWsIPaU1GWLEeZzC38Un99vLE0VW+GWq9wZ/SbBB0UxxIgppRdip1/w4AIwhmg8RDe18uVAKDCs5Vok6LGXHU8T+LWedOLD3oub9qM8f/zVNkCd9OkaTWaRwl6MInlSo4rPZJ1H14iFufIvS6PZ7Lku5wkgBF2h5gim77Nmg9YTL9lOKy/xrVZtNWHlhVRELgfPM=[/tex]知[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]在此新基下的矩阵即为上三角形[tex=7.643x1.571]oZ8QkEAR79j3V33cbTJnGHsjekgfJLnwkinbIC74hzD4apZJM3XPNHbgyCaNUyMB[/tex][tex=2.357x1.286]xMsTZlGeZvg6/+arSu/S1w==[/tex][tex=18.286x9.071]lcADbR/1KKfN309xpdR5dKt4llS4HBQKZx6agLoIdHjSAP3tMfNF261dOMtrQwSEHNBPSluIqzAr8TYMaq9vtqkDLQumd3xYT4RVtRZyZPDEeIE3SyA+I/XXA3n+etrrSOWS1lPgSa6iIMKsZ4BFiUkBHV7LfQsrTtJ7XqxhwZJxzhNaVWIasYFpnnDJNdGSdAkkkdanepLQfia9s3W4kJ8Y4Xq1JYjAW4Mugd+y/MMXdCaeDj3cuoFYYDw3aWTUWnoyNA1caA8nVZoSfuiOMJig1pAb/I/DcqVxr68s1BXdaRDUCpl+jvAslNQjaakywtv6o1Mr9eO/4WRBE4WeU7QHqAjbZALPe26B72Mp6Fn/23PbaxA4qtUc2Nnz/1YE2EGEBS6nybwxWaiAiBmPOQ==[/tex]即存在可逆矩阵[tex=3.571x1.286]pQkITt1/UhTRyhQMox/kYA==[/tex] ，使[tex=3.0x1.214]rN84CqmtCk5MRAP5g+8tJQ==[/tex]成上三角形．

举一反三

内容

0
主对角线上全是１的上三角矩阵称为特殊上三角矩阵．１）设[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是一对称矩阵，[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]为特殊上三角矩阵，而[tex=3.857x1.143]qPbOPa2llKVdw9JpP1uKuIR3ci6q7f1T05yD8ZrWl+k=[/tex]，证明：[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]与[tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex]的对应顺序主子式有相同的值；2）利用以上结果证明定理７的充分性．
1
设 [tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]是 [tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex] 矩阵,证明:存在一个 [tex=2.429x1.071]kaIcCzgC6SpeVVzRje1dYA==[/tex]非零矩阵 [tex=0.786x1.0]ri6gmnf1+J9dGqG5/1sV6A==[/tex] 使 [tex=3.357x1.0]qTT9ohZSoF+wT3IvQFgnLFrXH47FWW3xwhW8sfEIcnrDuDKcS2V13Iv41U8aG2/R[/tex] 的充分必要条件是 [tex=3.0x1.357]RRZ9zlAN4pWdGS7d9wHOkmrotL417Su2vM8Jrbh5h98=[/tex]
2
主对角线上全是１的上三角矩阵称为特殊上三角矩阵．证明：如果对称矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]的顺序主子式全不为零，那么一定有一个特殊上三角矩阵[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]使[tex=2.286x1.143]vB636PfIaAtueeIhKgyEvg==[/tex] 成对角形；
3
已知[tex=1.786x1.214]IENxQEh5u4RdnCaqHm72Xg==[/tex]为3阶矩阵，且[tex=6.5x1.357]Xw38Dcvrbs7IEKOZRvkd5g==[/tex]，其中[tex=0.786x1.0]XvHgf70VtK2FH5G93l0k3g==[/tex]是3阶单位矩阵．(1)证明：矩阵[tex=2.786x1.143]RcZ2ZRIlzxNTbD8lUHAX+Q==[/tex]可逆；(2)若[tex=7.786x3.5]DgXZT9CtCPAglTYwc4pEdVwGPrEvfplbNSz07f1CHm3lKZFzRkIi88nqRWCa7cdxtDn1Uq6Au4bDH+3NSK9+pGWuIrunnKgMXUiXxap7tYqS5e4P0ZLrWW76zZyDl/um[/tex]，求矩阵[tex=0.786x1.0]Yn3GgEZev6SOu2r4v1WnCw==[/tex]
4
设数域 [tex=0.643x1.0]0WA5oCO54gKWR/jKi5M2Zw==[/tex]上 [tex=0.643x0.786]/he/ol8BkDuTTL9yMPtH4Q==[/tex] 阶矩阵 [tex=0.786x1.0]b4HkKtHXeHofHX/gJc8Agg==[/tex] 的特征值 [tex=5.786x1.214]oNH2de8I1XfFs1vBi4Ose/m3xb4ZXIOWJL213dkS9oZGcEJxwIaoBVvUWo01TUpn[/tex] 全在 [tex=0.643x1.0]0WA5oCO54gKWR/jKi5M2Zw==[/tex] 中, 则存在 [tex=0.643x1.0]0WA5oCO54gKWR/jKi5M2Zw==[/tex] 上的可逆矩阵 [tex=0.643x1.0]WUJ/JHItsc3Bqx1WYNJcrg==[/tex], 使 [tex=3.143x1.214]Wy8xQjMsBEyjJUwCYAP+RQ==[/tex] 是上三角矩阵. 特别, 任一矩阵均复相似于某个上三角矩阵.