How does it follow that $A^T A = I$ from $m_{ij}m_{ik}=\delta _{jk}$?

Question

How does it follow that $A^TA = I$ from $m_{ij}m_{ik}$ = $\delta_{jk}$ (Where A is a nxn matrix).

Answer 1

Assuming $m_{ij}$ corresponds to the entries in $A$, it isn't so much that one follows from the other as the two expressions say the same thing written in two different ways. $m_{ij}m_{ik} =\delta_{jk}$ focuses on the separate entries and the actual calculations involved in the relationship and $A^TA=I$ focuses on the matrices as a whole, but other than that it's the same.

It's analogous (although, admittedly, not quite as immediate) to asking how $\vec x=\vec y$ follows from $x_i=y_i$.

Answer 2

$$\left(A^{T}A\right)_{i,j}=\sum_{k=0}^{n}m_{k,i}m_{k,j}=\sum_{k=0}^{n}\delta_{i,j}=\left(I\right)_{i,j}$$