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In this paper, we first present simple proofs of Chois results [4], then we give a short alternative proof for Fiedler and Markhams inequality [6]. We also obtain additional matrix inequalities related to partial determinants.
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A formula for the partial trace of a full symmetrizer is obtained. The formula is used to provide an inductive proof of the well-known formula for the dimension of a full symmetry class of tensors.
The trace regression model, a direct extension of the well-studied linear regression model, allows one to map matrices to real-valued outputs. We here introduce an even more general model, namely the partial-trace regression model, a family of linear mappings from matrix-valued inputs to matrix-valued outputs; this model subsumes the trace regression model and thus the linear regression model. Borrowing tools from quantum information theory, where partial trace operators have been extensively studied, we propose a framework for learning partial trace regression models from data by taking advantage of the so-called low-rank Kraus representation of completely positive maps. We show the relevance of our framework with synthetic and real-world experiments conducted for both i) matrix-to-matrix regression and ii) positive semidefinite matrix completion, two tasks which can be formulated as partial trace regression problems.
We show that ||u*u - v*v|| leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decomposition.
Let $A$ be a positive semidefinite $mtimes m$ block matrix with each block $n$-square, then the following determinantal inequality for partial traces holds [ (mathrm{tr} A)^{mn} - det(mathrm{tr}_2 A)^n ge bigl| det A - det(mathrm{tr}_1 A)^m bigr|, ] where $mathrm{tr}_1$ and $mathrm{tr}_2$ stand for the first and second partial trace, respectively. This result improves a recent result of Lin [14].
Let $A$ be an $mtimes m$ positive semidefinite block matrix with each block being $n$-square. We write $mathrm{tr}_1$ and $mathrm{tr}_2$ for the first and second partial trace, respectively. In this note, we prove the following inequality [ (mathrm{tr} A)I_{mn} - (mathrm{tr}_2 A) otimes I_n ge pm bigl( I_motimes (mathrm{tr}_1 A) -Abigr). ] This inequality is not only a generalization of Andos result [1], but it also could be regarded as a complement of a recent result of Choi [8]. Additionally, some new partial traces inequalities for positive semidefinite block matrices are also included.
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