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Консультация № 185981
07.05.2012, 02:34
0.00 руб.
07.05.2012, 08:26
0
1
1
Здравствуйте! Прошу помощи в следующем вопросе:
Обсуждение
07.05.2012, 09:20
общий
это ответ
Здравствуйте, Даша!
Собственные числа матрицы A являются решением характеристического уравнения
Корни этого уравнения - [$955$][sub]1[/sub] = -2, [$955$][sub]2[/sub] = 3, [$955$][sub]3[/sub] = 8. Собственный вектор X[sub]i[/sub], соответствующий собственному числу [$955$][sub]i[/sub], будет решением матричного уравнения (A - [$955$][sub]i[/sub]E)X = 0. Подставляя по очереди найденные собственные числа, решим соответствующие системы:
1) [$955$][sub]1[/sub] = -2
или
Решением будет x[sub]1[/sub] = x[sub]2[/sub] = t, x[sub]3[/sub] = 0.
2) [$955$][sub]2[/sub] = 3
или
Решением будет x[sub]1[/sub] = x[sub]2[/sub] = x[sub]3[/sub] = t.
3) [$955$][sub]3[/sub] = 8
или
Решением будет x[sub]1[/sub] = 0, x[sub]2[/sub] = x[sub]3[/sub] = t.
Здесь t - любое вещественное число, то есть собственный вектор определяется с точностью до постоянного коэффициента.
Итак, собственные числа и соответствующие собственные векторы линейного оператора A будут:
Собственные числа матрицы A являются решением характеристического уравнения
Корни этого уравнения - [$955$][sub]1[/sub] = -2, [$955$][sub]2[/sub] = 3, [$955$][sub]3[/sub] = 8. Собственный вектор X[sub]i[/sub], соответствующий собственному числу [$955$][sub]i[/sub], будет решением матричного уравнения (A - [$955$][sub]i[/sub]E)X = 0. Подставляя по очереди найденные собственные числа, решим соответствующие системы:
1) [$955$][sub]1[/sub] = -2
или
Решением будет x[sub]1[/sub] = x[sub]2[/sub] = t, x[sub]3[/sub] = 0.
2) [$955$][sub]2[/sub] = 3
или
Решением будет x[sub]1[/sub] = x[sub]2[/sub] = x[sub]3[/sub] = t.
3) [$955$][sub]3[/sub] = 8
или
Решением будет x[sub]1[/sub] = 0, x[sub]2[/sub] = x[sub]3[/sub] = t.
Здесь t - любое вещественное число, то есть собственный вектор определяется с точностью до постоянного коэффициента.
Итак, собственные числа и соответствующие собственные векторы линейного оператора A будут:
5
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