Positive square root and covariance matrices

Assume AC. Let $V$ be a nonzero finite-dimensional complex inner-product space and let $C$ be a positive covariance matrix, that is a self-adjoint operator on $V$ with $⟨Cx,x⟩\ge 0$ for every $x$ (Self-adjoint, positive, unitary and normal operators, Order on bounded self adjoint operators). Then $C$ has a unique positive square root $C^{1\mathrm{/}2}$, obtained by the same continuous functional calculus as on the main page: since $V$ is finite dimensional and nonzero the spectrum of $C$ is a finite nonempty subset of $[0,+\mathrm{\infty })$, and $\lambda \mapsto \sqrt{\lambda }$ is applied to $C$ by the calculus, the square root lying in $C^{\ast }(I,C)$ (Positive square root).

Unitary covariance change. If $U$ is unitary on $V$, then $(UC{U}^{\ast }{)}^{1\mathrm{/}2}=U{C}^{1\mathrm{/}2}{U}^{\ast }$: the operator $U{C}^{1\mathrm{/}2}{U}^{\ast }$ is positive, because $⟨U{C}^{1\mathrm{/}2}{U}^{\ast }x,x⟩=⟨C^{1\mathrm{/}2}{U}^{\ast }x,U^{\ast }x⟩\ge 0$, and its square is $U{C}^{1\mathrm{/}2}{U}^{\ast }U{C}^{1\mathrm{/}2}{U}^{\ast }=UC{U}^{\ast }$; uniqueness of the positive square root therefore identifies it with $(UC{U}^{\ast }{)}^{1\mathrm{/}2}$. In particular the square root is equivariant under the unitary changes of coordinates in which covariance matrices are compared.

Scope. This remark is orientation for the probability track: covariance matrices are positive, and the finite-dimensional instance of the positive-square-root theorem supplies their standard square roots. It proves no probability theorem, and it does not assert positivity of any particular covariance matrix; that positivity is a hypothesis of the interface, to be supplied by the consumer (The Axiom of Choice for the declared choice strength).