dipta007's Second 🧠

    Trace-Operator

    07-09-2022 || 17:32
    Tags: #linear-algebra

    trace-operator

    Trace operator is the sum of all the diagonal entries of a matrix

    $$
    Tr(A) = \sum_{i=1}^n A_{i, i}
    $$

    Trace operator gives another way of writing frobenius-norm of a matrix

    $$
    ||A||_F = \sqrt{Tr(AA^T)}
    $$

    Properties of Trace operator:

    1. Trace operator is same for the transpose matrix
      $$
      Tr(A) = Tr(A^T)
      $$
    2. In trace operator calculation, you can move the last factor to the first and the resulting value would be same. The resultant matrix should be square in that case
      $$
      Tr(ABC) = Tr(CAB) = Tr(BCA)
      $$
    3. The cyclic permutation holds even if the resulting matrix has different shape, like for $A^{nm}$ and $B^{mn}$,
      $$
      Tr(AB) = Tr(BA)
      $$
      even if $AB$ is a $nn$ matrix and $BA$ is a $mm$ matrix
    4. Trace of a scaler value is also a scaler
      $$
      Tr(a) = a
      $$

    References