## On graded QR decompositions of products of matrices

G. W. Stewart

### Abstract

This paper is concerned with the singular values and vectors of a product $M_{m}=A_{1}A_{2}\cdots A_{m}$ of matrices of order $n$. The chief difficulty with computing them directly from $M_{m}$ is that with increasing $m$ the ratio of the small to the large singular values of $M_{m}$ may fall below the rounding unit, so that the former are computed inaccurately. The solution proposed here is to compute recursively the factorization $M_{m} = QRP^T$, where $Q$ is orthogonal, $R$ is a graded upper triangular, and $P^T$ is a permutation.

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### Key words

QR~decomposition, singular value decomposition, graded matrix, matrix product.

65F30.