Inverse of the Square Wave Matrix

The ith row of A represents a ±1-valued “square wave” function of j with halfperiod i. Thus xA, where x is a (row) N -vector, gives a picture of the waveform that is a linear combination of square waves with weights in x. Similarly yA finds weights so that a given waveform y is a weighted linear combination of these square waves. This matrix has a surprisingly simple inverse. More generally, we can invert any matrix in which the first row is arbitrary, except that it cannot begin with 0, and the ith row repeats each term in the first row i times until it reaches the end of the row, thus spreading out the function with larger periods. The matrix A will thus have entries