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Консультация № 195833
12.06.2019, 10:47
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1
Здравствуйте! У меня возникли сложности с таким вопросом:
В трехмерном пространстве задана декартова прямоугольная система координат. Най-
ти собственные значения и собственные векторы оператора ортогонального отражения
относительно плоскости x + y = 0.
В трехмерном пространстве задана декартова прямоугольная система координат. Най-
ти собственные значения и собственные векторы оператора ортогонального отражения
относительно плоскости x + y = 0.
Обсуждение
14.06.2019, 10:25
общий
это ответ
Здравствуйте, m6996996!
195833
Я понимаю под ортогональным отражением относительно плоскости
или 
линейное преобразование трёхмерного пространства, переводящего вектор 
в вектор 
так, что
или, в матричной форме,
Следовательно,
-- матрица рассматриваемого оператора.
Составим характеристическое уравнение и решим его.
Корни характеристического уравнения (спектр матрицы
): 
Все корни вещественные числа; они являются собственными числами рассматриваемого оператора.
Пусть
Тогда
или
то есть
где 
-- первый собственный вектор.
Пусть
Тогда
или
то есть
где 
-- второй собственный вектор.
Литература
Апатенок Р. Ф. и др. Элементы линейной алгебры. -- Минск: Вышэйшая школа, 1977. -- 256 с.
195833
Я понимаю под ортогональным отражением относительно плоскости
или линейное преобразование трёхмерного пространства, переводящего вектор в вектор так, чтоили, в матричной форме,
Следовательно,
-- матрица рассматриваемого оператора.
Составим характеристическое уравнение и решим его.
Корни характеристического уравнения (спектр матрицы
): Все корни вещественные числа; они являются собственными числами рассматриваемого оператора.Пусть
Тогдаили
то есть
где -- первый собственный вектор.Пусть
Тогдаили
то есть
где -- второй собственный вектор.Литература
Апатенок Р. Ф. и др. Элементы линейной алгебры. -- Минск: Вышэйшая школа, 1977. -- 256 с.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
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