Inverse shift

What is the inverse of shifting a sequence to the right? Shifting it to the left, obviously. But wait a minute. Suppose you have a sequence of eight bits abcdefgh and you shift it to the right. You get 0abcdefg. If you shift this sequence to the left you get abcdefg0 You can’t recover the last element h because the right-shift destroyed information about h. A left-shift doesn’t fully recover a right-shift, and yet surely left shift and right shift are in some sense inverses. Yesterday I wrote a post about representing bit manipulations, including shifts, as matrix operators. The matrix corresponding to shifting right by k bits has 1s on the kth diagonal above the main diagonal and 0s everywhere else. For example, here is the matrix for shifting an 8-bit number right two bits. A black square represents a 1 and a white square represents a 0. This matrix isn’t invertible. When you’d like to take the inverse of a non-invertible matrix, your kneejerk response should be to compute the…