Trace

The trace of a square matrix is the sum of its diagonal entries:

\[\operatorname{tr}(\mathbf{A}) = \sum_{i=1}^n A_{ii}\]

The trace has several nice algebraic properties:

(i) \(\operatorname{tr}(\mathbf{A}+\mathbf{B}) = \operatorname{tr}(\mathbf{A}) + \operatorname{tr}(\mathbf{B})\)

(ii) \(\operatorname{tr}(\alpha\mathbf{A}) = \alpha\operatorname{tr}(\mathbf{A})\)

(iii) \(\operatorname{tr}(\mathbf{A}^{\!\top\!}) = \operatorname{tr}(\mathbf{A})\)

(iv) \(\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D}) = \operatorname{tr}(\mathbf{B}\mathbf{C}\mathbf{D}\mathbf{A}) = \operatorname{tr}(\mathbf{C}\mathbf{D}\mathbf{A}\mathbf{B}) = \operatorname{tr}(\mathbf{D}\mathbf{A}\mathbf{B}\mathbf{C})\)

The first three properties follow readily from the definition.

The last is known as invariance under cyclic permutations.

Note that the matrices cannot be reordered arbitrarily, for example \(\operatorname{tr}(\mathbf{A}\mathbf{B}\mathbf{C}\mathbf{D}) \neq \operatorname{tr}(\mathbf{B}\mathbf{A}\mathbf{C}\mathbf{D})\) in general.

Also, there is nothing special about the product of four matrices – analogous rules hold for more or fewer matrices.