representative matrix of a linear transformationgiven a linear transformation: T : M n ( C ) → M n ( C ), T ( A ) = A − 2 A T , what is the

Question

representative matrix of a linear transformationgiven a linear transformation:   T  :      M    n    (  C  )  →      M    n    (  C  ),   T  (  A  )  =  A  −  2      A    T  ,  what is the

Nicolas Calhoun · Accepted Answer

Let   A denote the subspace of symmetric (antisymmetric matrices). Denote by       E          i      j       the matrix all of whose coefficients are zero except in the entry at the intersection of the i-th line and the   j-th column, equal to 1. Thus (  (      E          i      j        )) is the canonical basis of   V.For any   i  &amp;lt;  j we have       s    p    a    n    (      E          i      j        ,      E          j      i        )  =      s    p    a    n    (      S          i      j        ,      A          i      j        ) where       S          i      j        =                    E                  i          j                    +              E                  j          i                      2   and       A          i      j        =                    E                  i          j                    −              E                  j          i                      2  The transpose map   τ  :  A  ↦      A      T   is easily seen to be diagonal in that basis : we have  τ      E          i      i        =      E          i      i        ,  τ      S          i      j        =      S          i      j        ,  τ      A          i      j        =  −      A          i      j      Note that   S is the eigenspace of   τ corresponding to the eigenvalue   1. So  {      E          i      i        |  1  ≤  i  ≤  n  }  ∪  {      S          i      j        |  1  ≤  i  &amp;lt;  j  ≤  n  } is a basis of   S, and we deduce       d    i    m    (  S  )  =            n      (      n      +      1      )        2  &amp;lt;brSimilarly   {      S          i      j        |  1  ≤  i  &amp;lt;  j  ≤  n  } is a basis of   A, and we deduce       d    i    m    (  A  )  =            n      (      n      −      1      )        2  For   A  ∈  S we have       A      T    =  A so   σ  (  A  )  =  −  A.For   A  ∈  A we have       A      T    =  −  A so   σ  (  A  )  =  5  ASo the characteristic polynomial       χ    A   of   A is      χ    A    =  (  X  +  1      )                  d        i        m            (          S      )        (  X  −  5      )                  d        i        m            (          A      )        =  (  X  +  1      )                    n        (        n        +        1        )            2        (  X  −  5      )                    n        (        n        +        1        )            2            χ    A    =  (  X  +  1      )                  d        i        m            (          S      )        (  X  −  5      )                  d        i        m            (          A      )        =  (  X  +  1      )                    n        (        n        +        1        )            2        (  X  −  5      )                    n        (        n        +        1        )            2

representative matrix of a linear transformation given a linear transformation: T : M

Answered question

Answer & Explanation

New Questions in Linear algebra