scholarly journals Eigenvalue inequalities for positive block matrices with the inradius of the numerical range

2022 ◽  
Author(s):  
Jean-Christophe Bourin ◽  
Eun-Young Lee
Keyword(s):  
Numerical Range ◽  
Norm Inequality ◽  
Operator Norm ◽  
Diagonal Block ◽  
Positive Matrix ◽  
Full Matrix ◽  
Block Matrices ◽  
The Third ◽  
Four Blocks ◽  
Second Eigenvalue

We prove the operator norm inequality, for a positive matrix partitioned into four blocks in [Formula: see text], [Formula: see text] where [Formula: see text] is the diameter of the largest possible disc in the numerical range of [Formula: see text]. This shows that the inradius [Formula: see text] satisfies [Formula: see text] Several eigenvalue inequalities are derived. In particular, if [Formula: see text] is a normal matrix whose spectrum lies in a disc of radius [Formula: see text], the third eigenvalue of the full matrix is bounded by the second eigenvalue of the sum of the diagonal block, [Formula: see text] We think that [Formula: see text] is optimal and we propose a conjecture related to a norm inequality of Hayashi.

scholarly journals NUMERICAL RANGE AND POSITIVE BLOCK MATRICES

2020 ◽  
pp. 1-9
Author(s):  
JEAN-CHRISTOPHE BOURIN ◽  
EUN-YOUNG LEE
Keyword(s):  
Numerical Range ◽  
Diagonal Block ◽  
Partial Trace ◽  
Full Matrix ◽  
Block Matrices ◽  
Positive Matrices ◽  
Four Blocks ◽  
Image Position

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$ , especially the distance $d$ from $0$ to $W(X)$ . A special consequence is an estimate, $$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$ between the diameters of the numerical ranges for the full matrix and its partial trace.

scholarly journals Some extremal algebras for hermitians

2001 ◽  
Vol 43 (1) ◽  
pp. 29-38
Author(s):  
M. J. Crabb ◽  
J. Duncan ◽  
C. M. McGregor
Keyword(s):  
Numerical Range ◽  
Banach Algebras ◽  
The Third

We study three extremal Banach algebras: (a) generated by two hermitian unitaries; (b) generated by an element of norm 1 all of whose odd positive powers are hermitian; (c) generated by an element of norm 1 all of whose even positive powers are hermitian. In all three cases the numerical range is found for various elements. The second algebra is shown to be isometrically isomorphic to a subalgebra of the first. The third algebra is identified with a space of functions.

Syntheses and X-ray structural analyses of [(C6H5)4As][Sn(EC6H5)3], E = S and Se

1985 ◽  
Vol 63 (2) ◽  
pp. 394-400 ◽  
Author(s):  
Philip A. W. Dean ◽  
Jagadese J. Vittal ◽  
Nicholas C. Payne
Keyword(s):  
Basal Plane ◽  
Nmr Spectra ◽  
Stannous Chloride ◽  
Monoclinic Space Group ◽  
Full Matrix ◽  
X Ray ◽  
Triangular Pyramid ◽  
The Third ◽  
Unit Cells ◽  
Least Squares Techniques

Stannous chloride, in solutions containing at least three mole equivalents of NaEC6H5, E = S and Se, forms triligated anions which crystallize readily as the tetraphenylarsonium salts, [(C6H5)4As[Sn(SC6H5)3], 1, and [(C6H5)4As][Sn(SeC6H5)3], 2. The crystals so formed are isomorphous, and their structures have been determined by single crystal X-ray diffractometry techniques. The salts crystallize in the monoclinic space group P21/c, with four formula units in unit cells of dimensions a = 10.760(1), b = 17.515(2), c = 20.247(3) Å, β = 104.71(1)° for 1, and a = 10.885(2), b = 17.587(2), c = 20.612(2) Å, β = 105.47(1)°, for 2. The structures have been refined by full-matrix least-squares techniques on F to agreement factors R = 0.024 (4780 observations with Fo > 3σ(Fo)) for 1, and R = 0.030 (3442 observations with Fo > 3σ(Fo)) for 2. The discrete [Sn(EC6H5)3]− anions are isostructural; the three S or Se atoms form the base of a triangular pyramid whose apex is occupied by the Sn atom. Two Sn—S distances in 1 are 2.532(1) and the third 2.552(1) Å, while in 2 the Sn—Se distances are 2.649(1), 2.650(1), and 2.671(1) Å. Angles at the Sn atom are 89.91(3), 90.55(3), and 96.73(3)° in 1, and 88.74(3), 89.72(3), and 97.27(3)° in 2. In both anions all three phenyl groups adopt a propeller-like conformation and are disposed in equatorial positions above the basal plane of chalcogen atoms. These salts represent the first thiolato- and sele-nolatostannates(II) to be characterized structurally. Tin-119 nmr spectra confirm that the [Sn(EC6H5)3]− ions are the same anionic Sn(II) species recently prepared insitu in solution.

scholarly journals Norm estimations, continuity, and compactness for Khatri-Rao products of Hilbert Space operators

2018 ◽  
Vol 14 (4) ◽  
pp. 382-386
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Hilbert Space ◽  
Trace Class ◽  
Compact Operators ◽  
Operator Norm ◽  
Trace Norm ◽  
Block Matrices ◽  
Hilbert Space Operators ◽  
Schmidt Norm

We provide estimations for the operator norm, the trace norm, and the Hilbert-Schmidt norm for Khatri-Rao products of Hilbert space operators. It follows that the Khatri-Rao product is continuous on norm ideals of compact operators equipped with the topologies induced by such norms. Moreover, if two operators are represented by block matrices in which each block is nonzero, then their Khatri-Rao product is compact if and only if both operators are compact. The Khatri-Rao product of two operators are trace-class (Hilbert-Schmidt class) if and only if each factor is trace-class (Hilbert-Schmidt class, respectively).

Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States

2004 ◽  
Vol 56 (1) ◽  
pp. 134-167 ◽  
Author(s):  
Chi-Kwong Li ◽  
Ahmed Ramzi Sourour
Keyword(s):  
Numerical Range ◽  
Linear Operators ◽  
Operator Norm ◽  
Unitary Matrix ◽  
Normed Algebra ◽  
Numerical Radius ◽  
Matrix Algebras ◽  
Linear Maps ◽  
The One ◽  
Image Position

AbstractEvery norm v on Cn induces two norm numerical ranges on the algebra Mn of all n × n complex matrices, the spatial numerical rangewhere vD is the norm dual to v, and the algebra numerical rangewhere is the set of states on the normed algebra Mn under the operator norm induced by v. For a symmetric norm v, we identify all linear maps on Mn that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e., linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if v is not the ℓ1, ℓ2, or ℓ∞ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e., of the formA ⟼ Q*AQ, where Q is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ℓ1 and the ℓ∞ norms, the results are quite different.

scholarly journals Crystal and molecular structures of two triosmium carbonyl cluster complexes containing ortho-metalated phenylphosphines (μ-H)Os3(CO)8L[PFcPh(C6H4)] (L = PFcPh2 or CO)

1992 ◽  
Vol 70 (8) ◽  
pp. 2215-2223 ◽  
Author(s):  
William R. Cullen ◽  
Steven J. Rettig ◽  
Tu Cai Zheng
Keyword(s):  
Heavy Atom ◽  
Phenyl Group ◽  
Molecular Structures ◽  
Carbonyl Cluster ◽  
Subsequent Formation ◽  
Full Matrix ◽  
Space Group ◽  
The Third ◽  
Crystal And Molecular Structures ◽  
Ortho Metalation

Pyrolysis of Os3(CO)11(PFcPh2), 7, and Os3(CO)10(PFcPh2)2, 8, (FcH = (Fe(η-C5H5)2 affords (μ-H)Os3(CO)9[PFcPh(C6H4)], 5, and (μ-H)Os3(CO)8(FcPPh2)[PFcPh(C6H4)], 6, respectively. The structures of 5 and 6 are analogous; they both contain a bridging hydride and a μ3-PFcPh(C6H4) moiety resulting from the ortho-metalation of one phenyl group and subsequent formation of a η2-bond from that metalated phenyl to the third osmium atom. Crystals of 5, are triclinic, a = 11.380(2), b = 15.930(3), c = 9.467(1) Å, α = 92.82(1)°, β = 100.32(1)°, 7 = 71.55(1)°, Z = 2, space group [Formula: see text], and those of 6•CH2Cl2, are also triclinic, a = 13.151(3), b = 17.177(4), c = 12.619(3) Å, α = 91.98(2)°, β = 104.18(2)°, γ = 67.72(2)°, Z = 2, space group [Formula: see text]. The structures were solved by heavy atom methods and were refined by full-matrix least-squares procedures to R = 0.028 and 0.026 by using 8080 and 9336 reflections with I ≥ 3σ(I), respectively.

ON MINIMAL SETS OF -MATRICES WHOSE PAIRWISE PRODUCTS FORM A BASIS FOR

2018 ◽  
Vol 98 (3) ◽  
pp. 402-413 ◽  
Author(s):  
W. E. LONGSTAFF
Keyword(s):  
Matrix Algebra ◽  
Full Matrix ◽  
Full Matrix Algebra ◽  
The Third ◽  
Minimal Sets

Three families of examples are given of sets of $(0,1)$-matrices whose pairwise products form a basis for the underlying full matrix algebra. In the first two families, the elements have rank at most two and some of the products can have multiple entries. In the third example, the matrices have equal rank $\!\sqrt{n}$ and all of the pairwise products are single-entried $(0,1)$-matrices.

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