How Can You Solve the Nonlinear System X'*X = A?

[sa_christina]

Hello all,
i would like to ask you how can be solved this system: X'*X=A, where A (pxp) known Matrix and X(nxp) the matrix i want to compute.

Thanks in advance

 

[Simon_Tyler]

I'm not sure of the best way to solve this but, if I'm interpreting what you wrote correctly, you can get an implicit solution like

$$\dot{X}^{\dagger }X=A<br /> \Longrightarrow \dot{X}^{\dagger }X+X^{\dagger }\dot{X}=A+A^{\dagger }$$
$$\Longrightarrow \partial _t\left(X^{\dagger }X\right)=A+A^{\dagger }$$
$$\Longrightarrow X^{\dagger }X=\int _0\left(A+A^{\dagger }\right)dt+X_0^{\dagger }X_0$$

 

[scisyhp2010]

$$X=U\Sigma V^T,X^T X=V\Sigma^T U^TU\Sigma V^T=V\Sigma^2 V^T=A$$

$$V\Sigma V^T\in X$$