What operation can we perform to convert an elliptical plot into a circular plot? Suppose, these are two pictures of plots of conditional probability density of a set of samples.

From the pictures, I can tell that,
(1) the left plot has non-zero covariance between the two dimensions(i.e. off diagonal elements are non-zero), and the means for each dimensions are below 0.
(2) On the right, the covariance matrix is diagonal(i.e. covariance=0. i.e. is spherical), mean=0.
Am I correct?
If YES, what mathematical operation can I apply to the left-plot to make it look like the right one?
 A: Without commenting on whether you are correct, I can tell you the following.
Let $\mathbf{X}$ be a $d$-dimensional random vector:
$$\mathbf{X}=(X_{1},X_{2},\ldots,X_{d})'$$
The joint distribution function can be expressed as:
$$F_{X}(x)=F_{X}(x_{1},\ldots,x_{d})=\text{Pr}(\mathbf{X}\leq \mathbf{x})=\text{Pr}(X_{1}\leq x_{1},\ldots,X_{d}\leq x_{d})$$
With marginal distribution of $X_{i}$:
$$F_{i}(x_{i})=\text{Pr}(X_{i}\leq x_{i})=F_{X}(\infty,\ldots,x_{i},\ldots,\infty)$$
We can say that $\mathbf{X}$ has an elliptical distribution if:
$$\mathbf{X}\overset{d}{=}\mathbf{\mu}+\mathbf{A}\mathbf{Y}$$
where $\mathbf{Y}\sim S_{k}(\psi)$, $\mathbf{A}\in \mathbb{R}^{d\times k}$ and $\mathbf{\mu}\in\mathbb{R}^{d}$. $S_{k}(\psi)$ means a $k$-dimensional spherical distribution with characteristic generator $\psi$.
Essentially, elliptical distributions are obtained via multivariate affine transformations of spherical distributions. This can be illustrated by the characteristic function $\phi_{X}$:
$$\phi_{X}(t)=\mathbb{E}\big(e^{it'\mathbf{X}}\big)=\mathbb{E}\big(e^{it'(\mathbf{\mu}+\mathbf{A}\mathbf{Y})}\big)=e^{it'\mathbf{\mu}}\mathbb{E}\big(e^{i(\mathbf{A}'t)'\mathbf{Y}}\big)=e^{it'\mathbf{\mu}}\psi(t'\mathbf{\Sigma} t)$$
where $\mathbf{\Sigma}=\mathbf{A}\mathbf{A}'$, which is a positive semidefinite matrix. The elliptical distribution is denoted by:
$$\mathbf{X}\sim E_{d}(\mathbf{\mu},\mathbf{\Sigma},\psi)$$
where $\mu$, $\Sigma$ and $\psi$ are referred to as the location vector, dispersion matrix and characteristic generator of the distribution, respectively. $E_{d}$ denotes a $d$-dimensional ellipitical distribution.
Thus, to transform from an elliptical distribution to a spherical one, the following transformation can be applied:
$$\mathbf{X}\sim E_{d}(\mathbf{\mu},\mathbf{\Sigma},\psi)\Longleftrightarrow \mathbf{\Sigma}^{-\tfrac{1}{2}}(\mathbf{X}-\mathbf{\mu})
\sim S_{d}(\psi)$$
This is analogous to the type of transformation you would do to a univariate normal random variable, $N(\mu,\sigma^{2})$, to obtain a standard normal random variable, $N(0,1)$:
$$Z\sim N(\mu,\sigma^{2})\Longleftrightarrow \sigma^{-1}(Z-\mu)
\sim N(0,1)$$
