trace of product of positive definite matrix

Question

For positive definite matrices $A$ and $C$, positive semidefinite matrices $B$ and $D$, I want to know whether $tr\{ABCD\}=0$ implies that $tr\{BD\}=0$.

Answer 1

No. Let $B=\begin{bmatrix}1&0\\0&0\end{bmatrix}$, $D=\begin{bmatrix}0&0\\0&1\end{bmatrix}$, and $A=C=\begin{bmatrix}2&1\\1&1\end{bmatrix}$. Then $\mathrm{tr}(BD)=0$, but $\mathrm{tr}(ABCD)=1$.

Answer 2

No. Let

$$A := {\rm I}_2 := \begin{bmatrix} 1 \\ & 1 \end{bmatrix}, \quad B := \begin{bmatrix} 1 \\ & 0 \end{bmatrix}, \quad C := \begin{bmatrix} 1 & -1 \\ -1 & 2 \end{bmatrix}, \quad D := \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}.$$

Then: $$ABCD = 0, \quad BD = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix},$$ so $$\mathop{\rm tr}(ABCD) = 0, \quad \mathop{\rm tr}(BD) = 1.$$