Is it possible to recover covariance matrix?

Question

I have the following equation:

$$\Sigma= C^T(A^TDA+\Gamma)C$$ where $C$ is a $n\times1$ matrix, $A$ is a $k\times1$ matrix, $D$ is a $k\times k$ matrix, and $\Gamma$ is a $n\times n$ diagonal matrix.

Is it possible to obtain the covariance matrix $D$ is I have all other parameters? What would be the step-by-step process for this?

I apologize if this is a very basic question and would appreciate detailed answers. Thanks!

Answer 1

The equation can be rewritten as $(AC)^TD(AC) = \Sigma - C^T\Gamma C$. So, basically, you are asking to solve an equation of the form $$ v^TDv=c\tag{1} $$ for a positive semidefinite and singular matrix $D$. Clearly, a solution exists only if $v=0$ and $c=0$ or $v\ne0$ and $c\ge0$.

When $v=0$ and $c=0$, every covariance matrix $D$ is a solution.

When $v\ne0$, the general solution is given by $D=Q\Lambda Q^T$, where $Q$ is any real orthogonal matrix and $\Lambda=\operatorname{diag}(\lambda_1,\ldots,\lambda_k)$ is a nonnegative diagonal matrix such that

• if we write $w=Q^Tv=(w_1,\ldots,w_k)^T$, then $\sum_i w_i^2\lambda_i=c$,
• at leat one of $\lambda_1,\ldots,\lambda_k$ is zero.

It follows that you cannot recover $D$ from $A,C,\Gamma$ and $\Sigma$ in general, because there are infinitely many possible candidates of $D$.