As it was made quite clear in the comments, you are not at the stage when you can ask a sensible question. Thus, I am treating your question as a reference request. The first issue is that there is no canonical computability model when one deals with real numbers (or subgroups of $SL(n, \mathbb R)$). There are several ways to deal with this:

1. You can work with matrices whose entries belong to a fixed number field. (I am not sure you know what this means since your background is in physics.) Then one can work with any of the standard notions of computability.

2. Work with the Real RAM or BSS computational model, see

Blum, Leonore; Cucker, Felipe; Shub, Michael; Smale, Steve, Complexity and real computation. Foreword by Richard M. Karp, New York, NY: Springer. xvi, 453 p. (1997). ZBL0948.68068.

3. Work with bit-computability model, see e.g.

Weihrauch, Klaus, Computable analysis. An introduction, Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer. x, 281 p. (2000). ZBL0956.68056.

Once you have some idea about questions which make sense when dealing with real computations, you can take a look here:

de Graaf, Willem Adriaan, Computation with linear algebraic groups, Monographs and Research Notes in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-2290-2/hbk; 978-1-4987-2291-9/ebook). xiv, 327 p. (2017). ZBL1518.14001.

As it was made quite clear in the comments, you are not at the stage where you can ask a sensible question. Thus, I am treating your question as a reference request. The first issue is that there is no canonical computability model when one deals with real numbers (or subgroups of $SL(n, \mathbb R)$). There are several ways to deal with this:

1. You can work with matrices whose entries belong to a fixed number field. (I am not sure you know what this means since your background is in physics.) Then one can work with any of the standard notions of computability.

2. Work with the Real RAM or BSS computational model, see

Blum, Leonore; Cucker, Felipe; Shub, Michael; Smale, Steve, Complexity and real computation. Foreword by Richard M. Karp, New York, NY: Springer. xvi, 453 p. (1997). ZBL0948.68068.

3. Work with the bit-computability model, see e.g.

Weihrauch, Klaus, Computable analysis. An introduction, Texts in Theoretical Computer Science. An EATCS Series. Berlin: Springer. x, 281 p. (2000). ZBL0956.68056.

Once you have some idea about questions which make sense when dealing with real computations, you can take a look here:

de Graaf, Willem Adriaan, Computation with linear algebraic groups, Monographs and Research Notes in Mathematics. Boca Raton, FL: CRC Press (ISBN 978-1-4987-2290-2/hbk; 978-1-4987-2291-9/ebook). xiv, 327 p. (2017). ZBL1518.14001.

Lastly, there are various software packages allowing you to perform computations in groups (written, say, in GAP by de Graaf, Holt and many others).