Householder Matrix QR Decomposition Implementation

In this article, we will discuss a special type of matrix known as the Householder matrix and explore how to perform its decomposition using QR decomposition algorithms. The QR decomposition method is a fundamental matrix factorization technique that decomposes any given matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). The algorithm implementation typically involves iterative Householder transformations, where each transformation zeroes out elements below the diagonal in one column at a time. Each Householder reflector can be constructed using vector normalization and outer product operations, with the key computation being the reflection vector v = a ± ||a||e₁, where a is the target column vector. The transformation matrix is then formed as H = I - 2vvᵀ/vᵀv. The process continues recursively on smaller submatrices until full triangularization is achieved. Throughout this discussion, we will delve deeper into these computational concepts to help you better understand and apply them in practical scenarios. Therefore, we will elaborate on these aspects in detail in the following sections, providing you with comprehensive insights into both the mathematical foundation and implementation methodology.