Let y=Ax. A is a matrix n by m and m>n. Also, x gets its values from a finite alphabet. Elements of A and x can be complex numbers. How can i show if the mapping from x to y is injective for given A and alphabet (beside a search method)?
For example, let A and the alphabet be
[1 0 1/sqrt2 1/sqrt2] [0 1 1/sqrt2 -1/sqrt2]
respectively. For this alphabet, x can be [1 1 -1 -1]'. Actually, there are 2^m=16 possible options for x, considering all permutations. Using this permutation set and A matrix, the operation will be one-to-one even if we map larger space to smaller space! Indeed, it is possible to test the injectivity with this scale. However, when we have larger matrices, search algorithm (i.e., testing all mappings) would take very long. If i can, i would like to verify it without a search algorithm.