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Консультация № 184942
24.12.2011, 14:29
57.98 руб.
24.12.2011, 18:30
0
1
1
Здравствуйте! У меня возникли сложности с таким вопросом:
Найти собственные значения и собственные векторы линейного преобразования,заданного в некотором базисе матрицей A.
Найти собственные значения и собственные векторы линейного преобразования,заданного в некотором базисе матрицей A.
Обсуждение
24.12.2011, 17:57
общий
это ответ
Здравствуйте, Максим!
Приравниваем определитель к нулю для нахождения собственных значений:
|1-k 8 23|
|0 5-k 7|=0
|0 3 1-k|
(1-k)(5-k)(1-k)-21(1-k)=0 -> (1-k)(k^2-6k-16)=0 -> (1-k)(k+2)(k-8)=0
Собственные значения -2, 1 и 8.
k=-2
3x1+8x2+23x3=0
7x2+7x3=0
3x2+3x3=0
Решение: х2=-х3; х1=-5х3
Собственный вектор (-5х;-х;х), где х -любое число
k=1
8x2+23x3=0
4x2+7x3=0
3x2=0
Решение: х2=0; х3=0; х1 - любое
Собственный вектор (х;0;0), где х -любое число
k=8
-7x1+8x2+23x3=0
-3x2+7x3=0
3x2-7x3=0
Решение: х2=7х3/3; х1=125х3/21
Собственный вектор (125x/21;7x/3;х), где х -любое число
Приравниваем определитель к нулю для нахождения собственных значений:
|1-k 8 23|
|0 5-k 7|=0
|0 3 1-k|
(1-k)(5-k)(1-k)-21(1-k)=0 -> (1-k)(k^2-6k-16)=0 -> (1-k)(k+2)(k-8)=0
Собственные значения -2, 1 и 8.
k=-2
3x1+8x2+23x3=0
7x2+7x3=0
3x2+3x3=0
Решение: х2=-х3; х1=-5х3
Собственный вектор (-5х;-х;х), где х -любое число
k=1
8x2+23x3=0
4x2+7x3=0
3x2=0
Решение: х2=0; х3=0; х1 - любое
Собственный вектор (х;0;0), где х -любое число
k=8
-7x1+8x2+23x3=0
-3x2+7x3=0
3x2-7x3=0
Решение: х2=7х3/3; х1=125х3/21
Собственный вектор (125x/21;7x/3;х), где х -любое число
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