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