struggling on the properties of the idempotent matrix, namely for any n × n matrix, A 2 = A .. The projection matrix defined by M = I n − A ( A T A ) − 1 A T is an idempotent matrix. The question is, for any given n × m ( n &gt; m) matrix B ,, do we have M = I n − A ( A T A ) − 1 A T = I n − A B ( B T A T A B ) − 1 B T A T , since A B is basically the linear transformation of matrix A .

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struggling on the properties of the idempotent matrix, namely for any   n  ×  n matrix,       A      2    =  A  .. The projection matrix defined by   M  =      I      n    −  A            (              A              T            A      )              −      1            A      T   is an idempotent matrix. The question is, for any given   n  ×  m (  n  &amp;gt;  m) matrix   B  ,, do we have                    M                            =                              I                  n                −        A                              (                          A                          T                        A            )                                −            1                                    A                  T                                                  =                              I                  n                −        A        B                              (                          B                          T                                      A                          T                        A            B            )                                −            1                                    B                  T                          A                  T                ,            since   A  B is basically the linear transformation of matrix   A  .

hildiadau0o · Accepted Answer

First, your projection matrix makes sense if   r  a  n  k    A  =  k  &amp;lt;  n. Otherwise,   M  =  0.  If so,   M maps to the   n  −  k-dimensional linear space spanned by the vectors       c    i   orthogonal to the columns of   A.Now, when you consider       M    ¯    =      I      n    −  A  B            (              B              T                    A              T            A      B      )              −      1            B      T        A      T  , this operator maps to the linear subspace orthogonal to the column space of   A  B. Here, you can have different options depending on the rank of   A  B.Let, for instance,   r  a  n  k    A  B  =  r  a  n  k    A. In this case, the column space of   A  B is the same as the column space of   A and   M  =      M    ¯  . But this will not be the case if   r  a  n  k    A  B  &amp;lt;  r  a  n  k    A.

struggling on the properties of the idempotent matrix, namely for any n × n

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