On the Explicit Expression of Chordal Metric between Generalized Singular Values of Grassmann Matrix Pairs with Applications

2018 ◽  
Vol 39 (4) ◽  
pp. 1547-1563 ◽  
Author(s):  
Weiwei Xu ◽  
Hong-Kui Pang ◽  
Wen Li ◽  
Xueping Huang ◽  
Wenjie Guo
Keyword(s):  
Explicit Expression ◽  
Singular Values ◽  
Chordal Metric ◽  
Generalized Singular Values

The quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function

1995 ◽  
Vol 125 (6) ◽  
pp. 1179-1192 ◽  
Author(s):  
Hervé Le Dret ◽  
Annie Raoult
Keyword(s):  
Explicit Expression ◽  
Energy Function ◽  
Singular Values ◽  
Convex Envelope ◽  
External Loading ◽  
Stored Energy ◽  
Quasiconvex Envelope

We give an explicit expression for the quasiconvex envelope of the Saint Venant–Kirchhoff stored energy function in terms of the singular values. This envelope is also the convex, polyconvex and rank 1 convex envelope of the Saint Venant–Kirchhoff stored energy function. Moreover, it coincides with the Saint Venant–Kirchhoff stored energy function itself on, and only on, the set of matrices whose singular values arranged in increasing order are located outside an ellipsoid. It vanishes on, and only on, the set of matrices whose singular values are less than 1. Consequently, a Saint Venant–Kirchhoff material can be compressed under zero external loading.

On Low Rank Approximation of Linear Operators in p-Norms and Some Algorithms

2016 ◽  
Vol 33 (04) ◽  
pp. 1650023
Author(s):  
Yang Liu
Keyword(s):  
Linear Operator ◽  
Sparse Matrix ◽  
Matrix Completion ◽  
Singular Values ◽  
Linear Operators ◽  
Low Rank ◽  
Minkowski Distance ◽  
Low Rank Approximation ◽  
Rank Approximation ◽  
Generalized Singular Values

In this paper, we study the optimal or best approximation of any linear operator by low rank linear operators, especially, any linear operator on the [Formula: see text]-space, [Formula: see text], under [Formula: see text] norm, or in Minkowski distance. Considering generalized singular values and using techniques from differential geometry, we extend the classical Schmidt–Mirsky theorem in the direction of the [Formula: see text]-norm of linear operators for some [Formula: see text] values. Also, we develop and provide algorithms for finding the solution to the low rank approximation problems in some nontrivial scenarios. The results can be applied to, in particular, matrix completion and sparse matrix recovery.

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