A numerically reliable solution for the squaring-down problem in system design
Applied Numerical Mathematics
Structured pseudospectra and structured sensitivity of eigenvalues
Journal of Computational and Applied Mathematics
Inertia and Rank Characterizations of Some Matrix Expressions
SIAM Journal on Matrix Analysis and Applications
Solvability conditions and general solution for mixed Sylvester equations
Automatica (Journal of IFAC)
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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.