Does the equality $\mathbf{x^TBy}=\mathbf{y^TB^Tx}$ always hold?

Question

Suppose we have three matrices $\mathbf{x,y}$, and $\mathbf{B}$. Where $\mathbf{x,y}$ are column vectors of length $n$ and $m$ respectively and size of $\mathbf{B}$ is $n\times m$. In this case $$\mathbf{x^TBy}$$ is a scalar quantity and similarly $$\mathbf{y^TB^Tx}$$ is also a scalar quantity. Is it always true that $$\mathbf{x^TBy}=\mathbf{y^TB^Tx}?$$ or there are some strict conditions under which this can be true?

Answer 1

Yes. The transpose of a scalar is equals to itself:
$$x^TBy=\sum_{i,j}x_{i}B_{ij}y_{j}$$ And $$y^TB^Tx=\sum_{i,j}y_i(B^T)_{ij}x_j=\sum_{i,j}y_iB_{ji}x_j=\sum_{i,j}x_jB_{ji}y_i$$ And they are the same.

Answer 2

Yes because for any matrices $A$ and $B$ such that the product $AB$ is defined, the product ${}^{\mathrm{t}\!}B\:{}^{\mathrm{t}\mkern-6mu}A$ is too and we have $${}^{\mathrm{t}\mkern-2mu}(\mkern-1muAB)={}^{\mathrm{t}\!}B\:{}^{\mathrm{t}\mkern-6mu}A.$$