This contribution reviews the external and the internal representations of linear time-invariant systems. This is done both in the time and the frequency domains. The realization problem is then discussed. Given the importance of norms in robust control and model reduction, the final part of this contribution is dedicated to the definition and computation of various norms. Again, the interplay ...

The first Section below extends to m × n matrices the results on orthogonality and projection we have previously seen for vectors. The Sections thereafter use these concepts to introduce the Singular Value Decomposition (SVD) of a matrix, the pseudo-inverse, and its use for the solution of linear systems.

We obtain the singular value decomposition of multi-companion matrices. We completely characterise the columns of the matrix U and give a simple formula for obtaining the columns of the other unitary matrix, V , from the columns of U . We also obtain necessary and sufficient conditions for the related matrix polynomial to be hyperbolic.

We show that the singular value decomposition with respect to a certain inner product in R gives the generalized singular value decomposition for two matrices with M columns and different sizes of rows, introduced recently to compare two sets of DNA microarrays of different organisms. 2000 Mathematical Subject Classification: 15A18, 92D10

Several mathematical methods are discussed in this paper, which are applied in image compression and restoration. Singular value decomposition (SVD) is used in compressing image. Conjugate gradients (CG) method and truncated Singular value decomposition (TSVD) regularization method are applied in image restoration. From the experience results we can see that those methods are effective in image...

In this paper, a system of linear matrix equations is considered. A new necessary and sufficient condition for the consistency of the equations is derived by means of the generalized singular-value decomposition, and the explicit representation of the general solution is provided. Keywords—Matrix equation, Generalized inverse, Generalized singular-value decomposition.

We present a variation of Paige's algorithm for computing the generalized singular value decomposition (GSVD) of two matrices A and B. There are two innovations. The rst is a new preprocessing step which reduces A and B to upper triangular forms satisfying certain rank conditions. The second is a new 2 by 2 triangular GSVD algorithm, which constitutes the inner loop of Paige's algorithm. We pre...

A data set with n measurements on p variables can be represented by an n × p data matrix X. In highdimensional settings where p is large, it is often desirable to work with a low-rank approximation to the data matrix. The most prevalent low-rank approximation is the singular value decomposition (SVD). Given X, an n × p data matrix, the SVD factorizes X as X = UDV ′, where U ∈ Rn×n and V ∈ Rp×p ...

In the present study, we demonstrate how to perform, using quantum annealing, the singular value decomposition and the principal component analysis. Quantum annealing gives a way to find a ground state of a system, while the singular value decomposition requires the maximum eigenstate. The key idea is to transform the sign of the final Hamiltonian, and the maximum eigenstate is obtained by quan...