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Let   A   be an    m  ×  m    complex matrix with zero trace. Then there are    m  ×  m    matrices   B   and   C   such that    A  =  [  B  ,  C  ]    and∥  B  ∥  ∥  C  ∥   2   ⩽    (  log  m  +  O   (  1  )   )    1  /  2∥  A  ∥   2where    ∥  D  ∥    is the norm of   D   as an operator on    ℓ  2  m    and∥  D  ∥   2    is the Hilbert–Schmidt norm of   D  . Moreover, the matrix   B   can be taken to be normal. Conversely, there is a zero trace    m  ×  m    matrix   A   such that whenever    A  =  [  B  ,  C  ]   ,∥  B  ∥  ∥  C  ∥   2   ⩾    |  log  m  −  O   (  1  )   |    1  /  2∥  A  ∥   2for some absolute constant    c  >  0   .

The Hilbert–Schmidt version of the commutator theorem for zero trace matrices