Аналитическая геометрия и линейная алгебра. Умнов А.Е. - 277 стр.

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МФТИ | Москва

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                                           a                          a                                    a                          a

                                                                          =    ( A −   A
                                                                                             a
                                                                                                 ) + a ( A −           A
                                                                                                                             a
                                                                                                                                 )a       =        

                                                                          =    ( A −   A
                                                                                             a
                                                                                                 )a ( A −         A
                                                                                                                        a
                                                                                                                             )a       ≥0 .
           
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                                               A =   ( A −     A       )2       = a   ( A −       A       )2a               =
                                                a                     a        a                           a
                                                                                                                                      
                                                    = a     ( ( A ) 2 − 2 A A a + ( A a ) 2 ) a                               =

                                                    = a    ( A ) 2 a − 2 A a a A a + ( A a ) 2 a a =
                                                                                                                                                      
                                                    = ( A ) 2 − 2 A A + ( A ) 2 = ( A ) 2 − ( A ) 2                                              .
                                                               a        a   a       a             a    a
           
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               A = ( A ) 2 − ( A                  a ( A ) 2 a − a A a
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                   a

                          = a A ( A a ) − a A a                    = a A ( λ a ) − a λ a                            = a λA a − a λ a
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                          = λ a λ a − λ2 a a                     = λ2 a a − λ2 a a                                 = λ2 − λ2 = 0 .
           
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