We will discuss some recent asymptotic results in matrix space and their extensions in Lie group, namely,
(1) Beurling-Gelfand-Yamamoto's theorem and the generalization of Huang and Tam.
(2) QR and Iwasawa asymptotic results of Huang and Tam.
(3) Francis-Kublanovskaya's QR algorithm and the generalization of Holmes, Huang and Tam.
(4) Rutishauer's LR algorithm and the generalization of Thompson and Tam.
These results are related to several important matrix decompositions, namely, SVD, QR decomposition, Gelfand-Naimark decomposition, Jordan decomposition and their counterparts, Cartan decomposition, Iwasawa decomposition, Bruhat decomposition, complete multiplicative Jordan decomposition