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The Frobenius (or Hilbert-Schmidt) norm of a matrix A is defined as the square root of the sum of the absolute value squared of all the elements of A:

||A||F = (sumisumj|aij|2)1/2

It can also be expressed as the square root of the sum of the norm of all the columns or rows of A. However, the more compact way to express the Frobenius norm is by noting that

||A||F = (tr ATA)1/2 = (tr AAT)1/2

where "tr" denotes the trace of matrix A, that is, the sum of its diagonal entries. If A contains complex elements, the hermitian conjugate of A should replace its transpose in the previous equation. This norm definition is to be compared to the most known p-norms, and of course satisfy the requirements common to all matrix norms:

  • ||A|| geq 0, and ||A|| = 0 only if A = 0;
  • ||A+B|| leq ||A||+||B||;
  • ||aA|| = |a| ||A||,
  • where "geq" and "leq" mean respectively "greater or equal than" and "lower or equal than", and a is an arbitrary constant.

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