# A Generalization of QR Factorization To Non-Euclidean Norms

I propose a way to use non-Euclidean norms to formulate a QR-like factorization which can unlock interesting and potentially useful properties of non-Euclidean norms - for example the ability of l^1 norm to suppresss outliers or promote sparsity. A classic QR factorization of a matrix ๐ computes an upper triangular matrix ๐ and orthogonal matrix ๐ such that ๐ = ๐๐. To generalize this factorization to a non-Euclidean norm ยท I relax the orthogonality requirement for ๐ and instead require it have condition number ฮบ ( ๐ ) = ๐ ^-1๐ that is bounded independently of ๐. I present the algorithm for computing ๐ and ๐ and prove that this algorithm results in ๐ with the desired properties. I also prove that this algorithm generalizes classic QR factorization in the sense that when the norm is chosen to be Euclidean: ยท=ยท_2 then ๐ is orthogonal. Finally I present numerical results confirming mathematical results with l^1 and l^โ norms. I supply Python code for experimentation.

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