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:
- Trace operator is same for the transpose matrix
$$
Tr(A) = Tr(A^T)
$$ - 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
$$
Tr(ABC) = Tr(CAB) = Tr(BCA)
$$ - 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 - Trace of a scaler value is also a scaler
$$
Tr(a) = a
$$