Solving Linear Equations AX=b Using QR Decomposition

In this problem, we will employ QR decomposition to solve the system of linear equations AX = b. QR decomposition is a matrix factorization technique that decomposes a matrix A into an orthogonal matrix Q and an upper triangular matrix R. This process can be efficiently implemented using Householder transformations, which are orthogonal transformations that zero out elements below the diagonal. The algorithm proceeds by applying a series of Householder reflections to transform the original matrix A into an upper triangular matrix R. The orthogonal matrix Q is implicitly stored as the product of these Householder vectors. Once decomposed, we solve the system by first computing Q^Tb (through backward accumulation of Householder vectors) and then solving the upper triangular system Rx = Q^Tb using back substitution. This method provides excellent numerical stability and is widely used in various applications including least squares regression, eigenvalue computations, and singular value decomposition. The Householder approach avoids the numerical instability that can occur with alternative methods like Gram-Schmidt orthogonalization.