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Консультация № 199790
06.12.2020, 18:58
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Здравствуйте! Прошу помощи в следующем вопросе:
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Обсуждение
13.12.2020, 18:30
общий
это ответ
Здравствуйте, marshal.bryus!
2. При приведении квадратичной формы
к нормальному виду методом Лагранжа возможны два случая. Если существует a[sub]ii[/sub][$8800$]0, производится замена
приводящая форму к виду
после чего для квадратичной формы Q[sub]i[/sub] (не содержащей переменной x[sub]i[/sub]) операция повторяется. Если же все a[sub]ii[/sub]=0, но существует a[sub]ij[/sub][$8800$]0, то производится замена x[sub]i[/sub] = y[sub]i[/sub]+y[sub]j[/sub], x[sub]j[/sub] = y[sub]i[/sub]-y[sub]j[/sub] (остальные переменные не меняются), сводящая форму к первому случаю.
В данном случае для формы
выполняем замену x[sub]1[/sub] = y[sub]1[/sub]+y[sub]2[/sub], x[sub]2[/sub] = y[sub]1[/sub]-y[sub]2[/sub], x[sub]3[/sub] = y[sub]3[/sub], приводящую форму к виду
для которого производим замену z[sub]1[/sub] = y[sub]1[/sub]+y[sub]3[/sub], z[sub]2[/sub] = y[sub]2[/sub], z[sub]3[/sub] = y[sub]3[/sub], приводящую форму к нормальному виду
С учётом того, что y[sub]1[/sub] = z[sub]1[/sub]-z[sub]3[/sub] получаем x[sub]1[/sub] = z[sub]1[/sub]+z[sub]2[/sub]-z[sub]3[/sub], x[sub]2[/sub] = z[sub]1[/sub]-z[sub]2[/sub]-z[sub]3[/sub], x[sub]3[/sub] = z[sub]3[/sub], то есть матрица линейного преобразования равна
2. При приведении квадратичной формы
к нормальному виду методом Лагранжа возможны два случая. Если существует a[sub]ii[/sub][$8800$]0, производится замена
приводящая форму к виду
после чего для квадратичной формы Q[sub]i[/sub] (не содержащей переменной x[sub]i[/sub]) операция повторяется. Если же все a[sub]ii[/sub]=0, но существует a[sub]ij[/sub][$8800$]0, то производится замена x[sub]i[/sub] = y[sub]i[/sub]+y[sub]j[/sub], x[sub]j[/sub] = y[sub]i[/sub]-y[sub]j[/sub] (остальные переменные не меняются), сводящая форму к первому случаю.
В данном случае для формы
выполняем замену x[sub]1[/sub] = y[sub]1[/sub]+y[sub]2[/sub], x[sub]2[/sub] = y[sub]1[/sub]-y[sub]2[/sub], x[sub]3[/sub] = y[sub]3[/sub], приводящую форму к виду
для которого производим замену z[sub]1[/sub] = y[sub]1[/sub]+y[sub]3[/sub], z[sub]2[/sub] = y[sub]2[/sub], z[sub]3[/sub] = y[sub]3[/sub], приводящую форму к нормальному виду
С учётом того, что y[sub]1[/sub] = z[sub]1[/sub]-z[sub]3[/sub] получаем x[sub]1[/sub] = z[sub]1[/sub]+z[sub]2[/sub]-z[sub]3[/sub], x[sub]2[/sub] = z[sub]1[/sub]-z[sub]2[/sub]-z[sub]3[/sub], x[sub]3[/sub] = z[sub]3[/sub], то есть матрица линейного преобразования равна
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