$A^{T}A$ is diagonal. What can I say about $A$?

Question

Is there any special property about the elements of $A$ if $A^{T}A$ is diagonal? I imagine you need some sort of symmetry but I can't see what it should be.

Edit: Sorry, maybe it's better phrased this way. If $A$ is $3 \times 3$, it has nine elements. If $A^{T}A$ is diagonal, how many parameters do I now need to specify $A$?

Answer 1

Concerning your second question: if $A$ is $3x3$ and satisfies $A^T A = D$ for some diagonal matrix $D$, it satisfies 6 additional constraints (namely the equations from the off-diagonal terms in $A^T A = D$). Since $A^T A$ is symmetric, only $3$ of these constraints are independent, so in total we need $9-3=6$ parameters to specify $A$.

Answer 2

The columns of $A$ are orthogonal to each other. So $A=OD$ where $O$ is an orthogonal matrix $OO^T=I$ and $D$ is a diagonal matrix.