1 Linear Algebra 1.5 Simultaneous Triangularization of Two Arbitrary Wide Matrices via the QR Decomposition 2 Probability Theory

1.6 Cyclic Shifts Property of Trace with Hadamard Products

The cyclic shift property of trace, i.e.,

\operatorname{trace}\left(\boldsymbol{W}_{1}\boldsymbol{W}_{2}\boldsymbol{W}_{% 3}\right)=\operatorname{trace}\left(\boldsymbol{W}_{3}\boldsymbol{W}_{1}% \boldsymbol{W}_{2}\right)=\operatorname{trace}\left(\boldsymbol{W}_{2}% \boldsymbol{W}_{3}\boldsymbol{W}_{1}\right),

holds for conventional matrix products of matrices $\boldsymbol{W}_{1},$ $\boldsymbol{W}_{2},$ and $\boldsymbol{W}_{3}.$ In the following theorem, we consider cyclic shifts of matrices involving Hadamard products, e.g., $\operatorname{trace}\left(\boldsymbol{W}_{1}\big{(}\boldsymbol{W}_{2}\circ% \boldsymbol{W}_{3}\big{)}\boldsymbol{W}_{4}\right).$

Theorem 5.

Let $m_{1},m_{2},m_{3}>0$ and let $\boldsymbol{W}_{1}\in\mathbb{C}^{m_{1}\times m_{2}},$ $\boldsymbol{W}_{2},\boldsymbol{W}_{3}\in\mathbb{C}^{m_{2}\times m_{3}},$ and $\boldsymbol{W}_{4}\in\mathbb{C}^{m_{3}\times m_{1}}$ be arbitrary matrices. Then,

\displaystyle\operatorname{trace}\left(\boldsymbol{W}_{1}\big{(}\boldsymbol{W}% _{2}\circ\boldsymbol{W}_{3}\big{)}\boldsymbol{W}_{4}\right)=\operatorname{% trace}\left(\big{(}\boldsymbol{W}_{2}^{\mathrm{T}}\circ(\boldsymbol{W}_{4}% \boldsymbol{W}_{1})\big{)}\boldsymbol{W}_{3}\right).

(1.8)

Proof.

We have

$\displaystyle\operatorname{trace}\left(\boldsymbol{W}_{1}(\boldsymbol{W}_{2}% \circ\boldsymbol{W}_{3})\boldsymbol{W}_{4}\right)$	$\displaystyle=\operatorname{trace}\left((\boldsymbol{W}_{2}\circ\boldsymbol{W}% _{3})\boldsymbol{W}_{4}\boldsymbol{W}_{1}\right)$
	$\displaystyle=\sum_{i=1}^{m_{2}}\sum_{j=1}^{m_{3}}[\boldsymbol{W}_{2}]_{i,j}[% \boldsymbol{W}_{3}]_{i,j}[\boldsymbol{W}_{4}\boldsymbol{W}_{1}]_{j,i}$
	$\displaystyle=\sum_{j=1}^{m_{3}}\sum_{i=1}^{m_{2}}[\boldsymbol{W}_{2}^{\mathrm% {T}}]_{j,i}[\boldsymbol{W}_{4}\boldsymbol{W}_{1}]_{j,i}[\boldsymbol{W}_{3}]_{i% ,j}$
	$\displaystyle=\operatorname{trace}\left(\big{(}\boldsymbol{W}_{2}^{\mathrm{T}}% \circ(\boldsymbol{W}_{4}\boldsymbol{W}_{1})\big{)}\boldsymbol{W}_{3}\right).$	(1.9)

∎

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First published: 18th May 2024