In the 2010 article "Closed-form Algorithms in Mobile Positioning: Myths and Misconceptions" by Niilo Sirola I came across the following expression: $$ x=(\Sigma^{-\frac{1}{2}}M)\ \setminus\ (\Sigma^{-\frac{1}{2}}v) $$ where $M$ is a $n$ by $3$ matrix and $v$ is a vector of length $n$.
In the article the author explains that the backslash operator \ is used as a shorthand for solving a linear system in least squares sense, i.e. $x = A\setminus b$ means "let x be the least-squares solution to $Ax = b$".
But he never explains the meaning of $\Sigma^{-1/2}$ and I wasn't able to google it or derive the meaning from context. By the looks of it, it should be a matrix operator that preserves the dimensions, since we need matrix on the left side of \ , and a column vector on the right side of it.
Is this a well-known notation? What does it mean?
This is the square root of a matrix, which given a symmetric, positive semi-definite matrix $A$ is the unique symmetric positive semi-definite matrix $B=A^{1/2}$ such that
$$A = B^⊤ B = B^2 $$
If you have an eigendecomposition $A=UDU^⊤$ with $D=\operatorname{diag}(λ_1, λ_2, …, λ_n)$, then $A^{1/2}$ is simply $UD^{1/2}U^⊤$ with $D^{1/2}=\operatorname{diag}(\sqrt{λ_1}, \sqrt{λ_2}, …, \sqrt{λ_n})$.
Taking the square root commutes with taking the inverse, so
$$ A^{-1/2} = (A^{1/2})^{-1} = (A^{-1})^{1/2}$$