I have a sparse lower triangular square matrix $L$. This is the Cholesky factor or the $R$ matrix from QR decomposition of a symmetric tridiagonal matrix $H_1$. Since $L$ is the Cholesky factor of a symmetric tridiagonal matrix $H_1$, it is sparse. I have another symmetric tridiagonal matrix $H_2$. I want to find the closest sparse lower triangular matrix $X$ such that: $L X^T \approx H_2$ (with the closeness being measured w.r.t Forbenius norm or in a least-squares sense). The matrix $H_2$ is not guaranteed to be positive definite and hence I cannot perform Cholesky decomposition or invert it. However $H_2$ is symmetric.
I tried using numpy.linalg.solve but this gives me a dense $X$ matrix. I also tried using numpy.linalg.solve_triangular, however, this gives me a dense lower triangular matrix.
If the question is ill-posed or if the information is inadequate, please do point it out.
I appreciate any recommendations. Thanks.