Proof: Uniqueness of the Dot Product Definition of the Matrix Transpose

Question

In section 3.2 of the textbook Linear Algebra and Its Applications, Fourth Edition by Gilbert Strang, the author introduces a dot product definition of the transpose matrix:

It is the matrix T such that $(Ax)^Ty = x^T(Ty)$ for vectors $x$ and $y$.

Using the alternative definition a transpose matrix as the original matrix with its rows and columns exchanged, it can be shown that the transpose matrix satisfies the above condition.

In proving the equivalence of the two definitions, what eludes me is the sufficiency of the condition. It doesn't seem immediately obvious to me that a single matrix satisfies this. Does anyone know of a proof for this?

Answer 1

If you let $A^\top$ denote the result of "exchanging the rows and columns" of $A$, then you can directly check that $(Ax)^\top = x^\top A^\top$ for any $x$. Thus $(Ax)^\top y = x^\top (A^\top y)$ holds for any $x,y$.

Conversely, suppose $(Ax)^\top y = x^\top (Ty)$ for all $x,y$. By considering $x=e_i$ and $y=e_j$ to be standard basis vectors, you immediately have $A_{ji} = T_{ij}$ for each $i,j$.

Answer 2

Hint: look at what the defining equation says if $x$ and $y$ are all the different possible pairs of basis vectors.