Find lower triangular matrix using Schur Factorization

Question

I need to find lower triangular matrix using Schur Factorization $A' = U^T A U$ . Actually after factorizing it results upper triangular matrix [using MATLAB]

Expecting result could be as such $$A'= \left[\matrix{a_{11} & 0 & 0 \\ a_{21} & a_{22} & 0 \\ a_{31} & a_{32} & a_{33}}\right]$$

Thanks

Answer 1

You may use Matlab to perform Schur factorization on $A^T$, and then transpose the upper triangular matrix to a lower triangular matrix. That is, suppose $R=U^TA^TU$ is upper triangular. Then $R^T=U^TAU$ is lower triangular.