QR factorization decomposes a full-column-rank data matrix X into the product of a matrix Q with mutually orthonormal columns (Q'Q = I) and an upper triangular matrix R, obtained by successively orthogonalizing and normalizing the columns of X via a Gram-Schmidt-like construction. In adaptive filtering, this factorization is reformulated with an exponentially-weighted (forgetting-factor) norm and recast as minimizing a norm-invariant quadratic cost under a unitary (norm-preserving) transformation Q(n), separating the error vector into an upper part solvable for the optimal combiner coefficients and a lower part representing the irreducible minimum error. Givens rotation is the specific unitary, angle-based operator used to compute this QR factorization recursively (time-updating Q(n) to Q(n+1)) rather than by brute-force recomputation, and belongs to numerical linear algebra as applied to recursive least-squares adaptive filter design, alongside alternative unitary techniques such as Householder transformation.
QR factorization decomposes a full-column-rank data matrix X into the product of a matrix Q with mutually orthonormal columns (Q'Q = I) and an upper triangular matrix R, obtained by successively orth…