I have a large matrix, around $10\times10$. Each individual element in the matrix itself is also a very large number, roughly of the order $10^{30}$.
I know that matrices can be used to solve linear equations. So if there is an equation $ax + by = c$ then can I denote the $10\times10$ matrix, $a$, using $x\text{ and }y$ which are just $2$ real numbers and $b\text{ and }c$ which are matrices. Can this be used to compress the larger matrix?
If not, then is there any other way that I can use so that I can use some form of short expression which can be evaluated into the larger matrix?
A $10\times10$ matrix is considered tiny and one can wonder whether it is worth compressing it.
A typical way to represent sparse matrices (i.e. with many zeroes) is to list the nonzero coefficients together with their indices. Several variants of this scheme are possible.
The sparse representation is more suitable for iterative resolution methods that essentially involve matrix/vector products. It is less amenable to direct resolution.