Let A be a matrix and Q be an orthogonal matrix such that A Q T is symmetric, positive semidefinite. Show that | | A + Q | | F ≥ | | A + P | | F for any orthogonal matrix P. Here, | | ⋅ | | F is the Frobenius norm.

Question

Let   A be a matrix and   Q be an orthogonal matrix such that   A      Q    T   is symmetric, positive semidefinite. Show that      |        |    A  +  Q      |              |        F    ≥      |        |    A  +  P      |              |        F  for any orthogonal matrix   P. Here,       |        |    ⋅      |              |        F   is the Frobenius norm.

nuvolor8 · Accepted Answer

Let   A      Q    T    =  V  D      V    T   be an orthogonal diagonalisation. Then   D is a nonnegative diagonal matrix and   U  =      V    T    P      Q    T    V is orthogonal. Since Frobenius norm is unitarily invariant, we have                    ‖        A        +        Q                  ‖          F          2                −        ‖        A        +        P                  ‖          F          2                                    =        ‖                  V          T                (        A        +        Q        )                  Q          T                V                  ‖          F          2                −        ‖                  V          T                (        A        +        P        )                  Q          T                V                  ‖          F          2                                                  =        ‖        D        +        I                  ‖          F          2                −        ‖        D        +        U                  ‖          F          2                                                  =        2        tr        ⁡        (        D        )        −        tr        ⁡        (        D        U        )        −        tr        ⁡        (                  U          T                D        )                                          =        2        tr        ⁡        (        D        (        I        −        U        )        )        ,            but this trace is nonnegative because   D  (  I  −  U  ) has a nonnegative diagonal.

Let A be a matrix and Q be an orthogonal matrix such that A Q T </m

Answered question

Answer & Explanation

New Questions in High school geometry