This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal.

The trace of the Kronecker product of two matrices is the product of their traces:

For example, consider the one-parameter family of linear transformations given by rotation through angle θ,

which clearly has trace zero, indicating that this matrix represents an infinitesimal transformation which preserves area.

The trace is a linear operator, hence it commutes with the derivative:

The trace also plays a central role in the distribution of quadratic forms.

For more properties and a generalization of the partial trace, see traced monoidal categories.

The operation of tensor contraction generalizes the trace to arbitrary tensors.