Vector structures and solutions of linear matrix equations

Vetter, W.

doi:10.1016/0024-3795(75)90010-5

Cited by 55 publications

(22 citation statements)

References 10 publications

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“…express the eigenvalues of K in terms of the eigenvalues of A and B, and this is one method that is considered to derive a condition for the invertibility of K. In [11] the equation Kx = q is converted to a dimension-reduced vector form, which is convenient for machine computations.…”

mentioning

confidence: 99%

The symmetric linear matrix equation

Ran

Reurings

2002

ELA

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The symmetric linear matrix equation

Ran

Reurings

2002

ELA

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“…Not all the elements in the vectors vec(Y) and vec(X) are independent, since the matrix arguments are symmetric. Of course this redundancy can be eliminated by using the symmetry condition to reduce the dimension of the vectors vec(Y) and vec(X) from their original m 2 and n 2 elements to only the m(m + 1)/2 and n(n + 1)/2 independent elements, respectively [27][28][29]. The corresponding rows of (CT X CT) are eliminated as well, reducing its size.…”

Section: Matrix-vector Representationmentioning

confidence: 99%

“…Here we present an outline for an algorithm to compute: q minE |Yi-CXCi Z (27) where _X= {x Ix> X > x} (28) defines the matrix interval set as described in Section 5. As shown in [21], the set X is the intersection of two cones and can be represented as the intersection of an infinite number of halfspaces:…”

Section: A Constrained Reconstruction Algorithmmentioning

confidence: 99%

Reconstructing Ellipsoids from Projections

Karl

Verghese

Willsky

1994

CVGIP: Graphical Models and Image Processing

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In this paper we examine the problem of reconstructing a (possibly dynamic) ellipsoid from its (possibly inconsistent) orthogonal silhouette projections. We present a particularly convenient representation of ellipsoids as elements of the vector space of symmetric matrices. The relationship between an ellipsoid and its orthogonal projections in this representation is linear, unlike the standard parameterization based on semi-axis length and orientation. This representation is used to completely and simply characterize the solutions to the reconstruction problem. The representation also allows the straightforward inclusion of geometric constraints on the reconstructed ellipsoid in the form of inner and outer bounds on recovered ellipsoid shape. The inclusion of a dynamic model with natural behavior, such as stretching, shrinking, and rotation, is similarly straightforward in this framework and results in the possibility of dynamic ellipsoid estimation. For example, the linear reconstruction of a dynamic ellipsoid from a single lower-dimensional projection observed over time is possible. Numerical examples are provided to illustrate these points.

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“…Bentler and Lee [1975] McDonald [1976] Vetter [1970] Hartwig [1972] (incorrect) Vetter [1973Vetter [ , 1975 MacRae [ 1974] Hartwig and Morris [1975] Swain [1975] Balestra [1976] Brewer [1977,1978] Magnus and Neudecker [1979] Barnett [1973] Henderson and Searle [1979] Henderson and Searle (this paper) …”

Section: Mcdonald and Swaminathan [1973]mentioning

confidence: 99%

“…(where dots represent zeros), are every third row, namely rows 1, 4, 2, 5, 3 and 6, respectively, of ;s. In contrast to operational definitions like (5) and (7) through ( (14) Vetter's [1973Vetter's [ , 1975 [1975]. Balestra [1976, p. Vetter [1973,1975] presents (17) as does Brewer [1977,1978],…”

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confidence: 99%