Direct Eigenvalue Reordering in a Product of Matrices in Periodic Schur Form

2006 ◽  
Vol 28 (1) ◽  
pp. 285-300 ◽  
Author(s):  
Robert Granat ◽  
Bo Kågström
Keyword(s):  
Schur Form ◽  
Product Of Matrices

scholarly journals Jordan products of quantum channels and their compatibility

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark Girard ◽  
Martin Plávala ◽  
Jamie Sikora
Keyword(s):  
Quantum State ◽  
Sufficient Conditions ◽  
The Other ◽  
Independent Interest ◽  
Quantum Channels ◽  
Semidefinite Programs ◽  
Jordan Product ◽  
Marginal Problem ◽  
Jordan Products ◽  
Product Of Matrices

AbstractGiven two quantum channels, we examine the task of determining whether they are compatible—meaning that one can perform both channels simultaneously but, in the future, choose exactly one channel whose output is desired (while forfeiting the output of the other channel). Here, we present several results concerning this task. First, we show it is equivalent to the quantum state marginal problem, i.e., every quantum state marginal problem can be recast as the compatibility of two channels, and vice versa. Second, we show that compatible measure-and-prepare channels (i.e., entanglement-breaking channels) do not necessarily have a measure-and-prepare compatibilizing channel. Third, we extend the notion of the Jordan product of matrices to quantum channels and present sufficient conditions for channel compatibility. These Jordan products and their generalizations might be of independent interest. Last, we formulate the different notions of compatibility as semidefinite programs and numerically test when families of partially dephasing-depolarizing channels are compatible.

Representations for the image-kernel (p,q)-inverses of block matrices in rings

2020 ◽  
Vol 27 (2) ◽  
pp. 297-305
Author(s):  
Dijana Mosić
Keyword(s):  
Block Matrix ◽  
Block Matrices ◽  
Schur Form

AbstractWe present the conditions for a block matrix of a ring to have the image-kernel{(p,q)}-inverse in the generalized Banachiewicz–Schur form. We give representations for the image-kernel inverses of the sum and the product of two block matrices. Some characterizations of the image-kernel{(p,q)}-inverse in a ring with involution are investigated too.

scholarly journals Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

2022 ◽  
Vol 48 (1) ◽  
pp. 1-36
Author(s):  
Mirko Myllykoski
Keyword(s):  
Eigenvalue Problems ◽  
Critical Path ◽  
Hessenberg Matrix ◽  
Single Node ◽  
Standard Case ◽  
Qr Algorithm ◽  
Schur Form ◽  
Tightly Coupled ◽  
Qz Algorithm ◽  
New Ideas

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

scholarly journals Swapping 2 × 2 blocks in the Schur and generalized Schur form

2020 ◽  
Vol 373 ◽  
pp. 112274
Author(s):  
Daan Camps ◽  
Nicola Mastronardi ◽  
Raf Vandebril ◽  
Paul Van Dooren
Keyword(s):  
Schur Form
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