On the Computation of the Restricted Singular Value Decomposition via the Cosine-Sine Decomposition
On the Computation of the Restricted Singular Value Decomposition via the Cosine-Sine Decomposition
复制标题
DOI：
10.1137/s0895479898346983
复制
发表时间：
2000-05
期刊：
影响因子：
--
通讯作者：
D. Chu;L. D. Lathauwer;B. Moor
中科院分区：
文献类型：
--
作者：
D. Chu;L. D. Lathauwer;B. Moor
本文证明了矩阵三元组$ a \ In \R^{n \ m}, B\ In \R^{n \ l}, C\ In \R^{p \ m}$的限制奇异值分解可以用余弦-正弦分解来计算。首先，将矩阵A， B， C简化为一个低维矩阵三元组${\cal A}, {\cal B}, {\cal C}$，其中${\cal B}$和${\cal C}$是非奇异的，利用列旋转的qr分解和URV分解等正交变换。第二步，由${\cal B}^{-1}{\cal A}{\cal C}^{-1}$的奇异值分解导出A、B、C的受限奇异值分解分量。而不是显式形成后一种产品，与余弦-正弦分解的联系，可以计算Van Loan的方法，被利用。数值算例表明了该方法的有效性。
In this paper, we show that the restricted singular value decomposition of a matrix triplet $A\in \R^{n \times m}, B\in \R^{n \times l}, C\in \R^{p \times m}$ can be computed by means of the cosine-sine decomposition. In the first step, the matrices A, B, C are reduced to a lower-dimensional matrix triplet ${\cal A}, {\cal B}, {\cal C}$, in which ${\cal B}$ and ${\cal C}$ are nonsingular, using orthogonal transformations such as the QR-factorization with column pivoting and the URV decomposition. In the second step, the components of the restricted singular value decomposition of A, B, C are derived from the singular value decomposition of ${\cal B}^{-1}{\cal A}{\cal C}^{-1}$. Instead of explicitly forming the latter product, a link with the cosine-sine decomposition, which can be computed by Van Loan's method, is exploited. Some numerical examples are given to show the performance of the presented method.