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Some norm inequalities for special Gram matrices

100%
Türkmen R., Kan O., Gökbas H.
Special Matrices
|
2016
|
tom 4
|
nr 1
262-269
EN
In this paper we firstly give majorization relations between the vectors Fn = {f0, f1, . . . , fn−1},Ln = {l0, l1, . . . , ln−1} and Pn = {p0, p1, . . . , pn−1} which constructed with fibonacci, lucas and pell numbers. Then we give upper and lower bounds for determinants, Euclidean norms and Spectral norms of Gram matrices GF=〈Fn,Fni〉, GL=〈Ln,Lni〉, GP=〈Pn,Pni〉, GFL=〈Fn,Lni〉, GFP=〈Fn,Pni〉.
2
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Embedded Lattice and Properties of Gram Matrix

100%
Futa Y., Shidama Y.
Formalized Mathematics
|
2017
|
tom 25
|
nr 1
73-86
EN
In this article, we formalize in Mizar [14] the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm [16] and cryptographic systems with lattice [17].
3
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Lattice of ℤ-module

100%
Futa Y., Shidama Y.
Formalized Mathematics
|
2016
|
tom 24
|
nr 1
49-68
EN
In this article, we formalize the definition of lattice of ℤ-module and its properties in the Mizar system [5].We formally prove that scalar products in lattices are bilinear forms over the field of real numbers ℝ. We also formalize the definitions of positive definite and integral lattices and their properties. Lattice of ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [14], and cryptographic systems with lattices [15] and coding theory [9].
4
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Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space

100%
Appi Reddy K., Kurmayya T.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using the indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.
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