# 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
$$Tr(a) = a$$