Whether ${\rm trace}(\mathbf{x}\mathbf{x}^TM)={\rm trace}(\mathbf{x}^TM\mathbf{x})$

Question

If $\mathbf{x} \in R^n,M \in R^{n \times n}$,and trace(.) is the sum of the diagonal elements of a matrix,whether the above conclusion is hold,if so,how to prove,thanks in advance

Answer 1

Correct. More generally, the trace is invariant to cyclic permutations:

$$Tr(ABC)=Tr(BCA)=Tr(CAB).$$

You can prove this using the regular trace identity

$$Tr(XY)=Tr(YX),$$

and let $X=AB,Y=C$ or let $X=A,Y=BC.$