When does $A^{-1}=A^T$ holds true?

Question

Is it true that if $D$ is a diagonal matrix and $AXA^T=D$ then $A^{-1}=A^T$?

Referring to my recent question which I struggle to find an answer to Help With Matrices Whitening Problem in Random Vectors

EDIT: |D| > 0 of course.

Answer 1

It isn't if you take $A = 2Id$ and $X = Id$ You have that $AXA^{T} = 4Id = D$ a diagonal matrix, but $A^{-1} \not= A^{T}$

Answer 2

No. Suppose that both $A$ and $D$ are the null matrix. Then $D$ is diagonal and $AXA^T=D$ (whatever $X$ is), but $A$ has no inverse.

Answer 3

For example

$$ \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)\left( \begin{array}{cc} 3 & -2 \\ -2 & 2 \\ \end{array} \right)\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 0 \\ 0 & 2 \\ \end{array} \right) $$

Just from solving

$$\left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right)\left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right) = \left( \begin{array}{cc} a+b+c+d & b+d \\ c+d & d \\ \end{array} \right) = D $$