 Solution

Let $r=\sqrt{x_i^2+x_j^2}$. If $r>0$, choose $\theta$ so thatSuch an angle exists because the two ratios have squared sum one. Then the changed coordinates areAll other coordinates satisfy $y_k=x_k$. If $r=0$, both input coordinates are zero; take $\theta =0$, giving the identity matrix on this pair. In all cases,This fixes a nonnegative retained coordinate and handles division by zero explicitly. For floating-point computation, the stable construction of a Givens rotation uses scaled Euclidean norm evaluation to avoid unnecessarily overflowing or underflowing the squares.