Problem in proving \begin{eqnarray*} \left \lVert Ax-y\right\rVert^{2} &=& x^{T} A^{T} A x -2 y^{T} A x + y^{T} y \end{eqnarray*}

Question

I begin with doing \begin{eqnarray*} \left \lVert Ax-y\right\rVert^{2} &=& (Ax-y)^{T}(Ax-y) \end{eqnarray*} \begin{eqnarray*} \left \lVert Ax-y\right\rVert^{2} &=& (x^{T}A^{T}-y^{T})(Ax-y) \end{eqnarray*}

\begin{eqnarray*} \left \lVert Ax-y\right\rVert^{2} &=& x^{T} A^{T} A x - y^{T} A x -x^{T} A^{T} y+ y^{T} y \end{eqnarray*}

So my doubt is how to show \begin{eqnarray*} \ y^{T} A x=x^{T} A^{T} y \end{eqnarray*}

because if it is true my proof will be completed.

Answer 1

$y^t Ax$ is just the dot product between $Ax$ and $y$, regarded as column vectors. Also, $x^tA^t y = (Ax)^t y$ is the dot product between $y$ and $Ax$.