Krdytkyu

$newcommand{psym}{text{Psym}_n}$
$newcommand{sym}{text{sym}}$
$newcommand{Sym}{operatorname{Sym}}$
$newcommand{Skew}{operatorname{Skew}}$
$renewcommand{skew}{operatorname{skew}}$
$newcommand{GLp}{operatorname{GL}_n^+}$

Denote by $psym$ the space of symmetric positive-definite $n times n$ matrices, and by $GLp$ the group of real $n times n$ invertible matrices with positive determinant.

Let $P:GLp to psym$ map each matrix $A$ to its unique positive factor in the polar decomposition, i.e. $P(A)=sqrt{A^TA}$.

I am trying to find a nice "closed-form algebraic expression" for the differential $dP_A$, where $A in psym$ is symmetric positive-definite. (So I am fine with using positive square roots, but not integral formulas or vectorization operations like here or here).

In other words: I want to find a formula for $dP_A(B)$, where $A$ is positive-definite and $B$ is an arbitrary matrix. Here is a partial result:

$dP_A(B)=operatorname{sym}(B) iff B in V_P:={Bin M_n , | , BP + P B^T=B^T P+ P B}$.

Proof: Differentiating $P^2=A^TA$ we get $ dot PP + P dot P = B^TA + A^TB.$ Since we assumed $A in psym$, we have $A=P$ at time $t=0$, so our equation becomes

$$ dot PP + P dot P = B^TP + PB,$$

and $dot P$ is the unique solution of this equation. Now it is easy to verify that $dot P=operatorname{sym}(B)$ is a solution if and only if:

$$ frac{B+B^T}{2}P + P frac{B+B^T}{2} = P B + B^T P iff BP + P B^T=B^T P+ P B iff B in V_p $$

Note that the presence of the "symmetrization operator" is natural here; $B to dP_A(B)$ is something which eats arbitrary matrices and returns symmetric matrices. (We also have the special case where $B$ is symmetric, and then $dP_A(B)=B$; this is also immediate from the fact that $P_{psym}=Id_{psym}$ and $B in T_A{psym}$).

The result mentioned above does not cover all the cases, since in general $V_p subsetneq M_n$.

Unfortunately, I couldn't come up with an expression for the general case. Since $dP_A$ is linear, and we always have $sym subseteq V_P$, it suffices to consider skew-symmetric matrices $B$ as inputs.

However, I don't see a clear pattern fur such matrices. In the simplest case of $n=2$, and

$$A=begin{pmatrix} sigma_1 & 0 \ 0 & sigma_2 end{pmatrix}, B=begin{pmatrix} 0 & a \ -a & 0 end{pmatrix}, , , text{ where } , , sigma_1 ge sigma_2 $$ a short calculation shows that

$$ dot P=dP_A(B)=begin{pmatrix} 0 & afrac{sigma_1-sigma_2}{sigma_1+sigma_2} \ afrac{sigma_1-sigma_2}{sigma_1+sigma_2} & 0 end{pmatrix}=frac{sigma_1-sigma_2}{sigma_1+sigma_2}begin{pmatrix} 0 & a \ a & 0 end{pmatrix}.$$

So, the map $B to dP_A(B)$, considered as a map $text{skew} to sym$ is of the form $begin{pmatrix} 0 & a \ -a & 0 end{pmatrix} to c(A)begin{pmatrix} 0 & a \ a & 0 end{pmatrix}$, where $C(A)=frac{sigma_1-sigma_2}{sigma_1+sigma_2}=sqrt{1-4frac{det A}{(text{tr}A)^2}}$ is a constant depending only on $A$.

I don't see an immediate way to write $begin{pmatrix} 0 & a \ -a & 0 end{pmatrix} to begin{pmatrix} 0 & a \ a & 0 end{pmatrix}$ in "algebraic way", i.e. involving only matrix addition, multiplication, and square roots.