Norm-preserving dilation theorems for a block positive semidefinite (definite) matrix

2021 ◽  
pp. 1-8
Author(s):  
Xifu Liu
Keyword(s):  
Positive Semidefinite

scholarly journals Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

2021 ◽  
Author(s):  
Stefano Massei
Keyword(s):  
Computer Science ◽  
Recent Work ◽  
Linear Algebra ◽  
Optimization Problems ◽  
Approximation Error ◽  
Positive Semidefinite ◽  
Numerical Linear Algebra ◽  
Maximum Volume ◽  
Deterministic Algorithms ◽  
Principal Submatrix

AbstractVarious applications in numerical linear algebra and computer science are related to selecting the $$r\times r$$ r × r submatrix of maximum volume contained in a given matrix $$A\in \mathbb R^{n\times n}$$ A ∈ R n × n . We propose a new greedy algorithm of cost $$\mathcal O(n)$$ O ( n ) , for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $$(r+1)$$ ( r + 1 ) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost $$\mathcal O(n^3)$$ O ( n 3 ) .

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

scholarly journals Rational and real positive semidefinite rank can be different

2016 ◽  
Vol 44 (1) ◽  
pp. 59-60 ◽  
Author(s):  
Hamza Fawzi ◽  
João Gouveia ◽  
Richard Z. Robinson
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Rank
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