Orthonormal columns implies orthonormal rows I find it non-intuitive if I impose that all of a square matrix's columns are normalized and mutually orthogonal, then all its rows are also normalized and mutually orthogonal. Any intuitive explanation for this? Also if I relax the conditions to be only mutually orthogonal without being normalized, is this still true? And why so?
 A: I assume you mean orthonormal coulmn implies orthonormal rows. If so, than:
Let $Q$ be a square matrix with orthonormal columns. Therefore we know:
$$Q'Q=I \implies Q'=q^{-1}$$
$$(Q')'Q'=QQ'=I$$
Therefore, we have that the rows are also orthonormal. 
A: The derivation done by @eminem is mathematically incorrect. ′=  does not imply (′)′′=. Below is a solution that could answer the question
$Q^\top Q=I => (Q^\top Q)^{-1}=I^{-1} => Q^{-1}Q^{\top -1}=I => QQ^{-1}Q^{\top -1}=Q => QQ^{-1}Q^{\top -1}Q^\top =QQ^\top => I=QQ^\top$
This means that orthonormal rows/columns imply orthonormal columns/rows. Of course, here I assume that $Q$ is a full rank square matrix. This conclusion is not always true for rectangle matrices.
A: For a matrix $Q$ with normalized orthogonal columns, $Q^TQ=I,$ then $Q^T=Q^{-1}$ and $QQ^T=Q Q^{-1}=I.$ $Q$ also has normalized orthogonal rows.
$A=\begin{bmatrix}1&0&2\\
1&-1&-1\\
1&1&-1\end{bmatrix}$ clearly is a matrix with orthogonal columns, but without orthogonal rows.
Assume $A^TA=D,$ $D$ is an invertible diagonal matrix. Let $AP=B,$ $P$ is a diagonal matrix which normalized columns. So $B$ is an orthogonal matrix. Let's see how $AP$'s rows are also normalized, i.e. $AP=B=QA,$ how $Q$ looks like?
$$PDP=P^TA^TAP=B^TB=I\Rightarrow P^2=D^{-1}\quad (\text{normalization, of course})$$
$$QAA^TQ^T=BB^T=I \Rightarrow Q^TQ=(AA^T)^{-1}$$
So $Q$ may not be a diagonal matrix. Otherwise $AA^T$ is also diagonal for sure, which contradicts our counter-example. This means when we normalize our column spaces, row basis may change during the operation.
I don't know what we can get further. I remember that when I study linear algebra, I don't find strong connection between row spaces and column spaces, especially geometrical meaning. I hope someone has learned matrix theory can answer this question deeply.
