Properties of a Unitary Matrix Obtained from a Sequence of Normalized Vectors

• Published in: SIAM journal on matrix analysis and applications Vol. 35; no. 2; pp. 526 - 545
• Main Authors: Paige, Christopher C, Wulling, Wolfgang
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
• Subjects: Algorithms; Decomposition; Error analysis; Iterative methods; Mathematical analysis; Norms; Orthogonality; Vectors (mathematics)
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/120897687

In [SIAM J. Matrix Anal. Appl., 31 (2009), pp. 565--583] it was shown how a special $(n\!+\!k)\times (n\!+\!k)$ unitary matrix can be defined from any sequence of $k$ vectors in $\mathbb{C}^n$ having unit Euclidean norms. This unitary matrix can be called an augmented orthogonal matrix when applied in the analysis of any algorithm that seeks to compute $k$ orthonormal $n$-vectors, but where the computed, then theoretically normalized, vectors $v_j$ in $V_k=[v_1,\ldots,v_k]$ have a significant loss of orthogonality. These unitary matrices can occur in other situations, being in fact products of $k$ particular Householder matrices (unitary elementary Hermitians), and they have many interesting theoretical properties. Several new results concerning them have been collected here so that they can be easily referenced, our main purpose being to facilitate the rounding error analyses of iterative orthogonalization algorithms which lose significant orthogonality, such as the Lanczos process and its many related procedures. A key component of the analysis is the $k\times k$ strictly upper triangular matrix $S_k$ arising from $V_k$. The singular value decomposition of $S_k$ reveals the CS decomposition of the $(n\!+\!k)\times (n\!+\!k)$ unitary matrix, the null space of $V_k$, and properties of the orthogonality and loss of orthogonality resulting from its columns. Among other things these properties are used to analyze the passage towards a complete set of orthonormal vectors in $\mathbb{C}^n$ and the contribution to orthogonality of any subsequent unit norm vector $v_{k+1}$. [PUBLICATION ABSTRACT]