The minimality of determinantal varieties

AbstractThe determinantal variety {\Sigma_{pq}} is defined to be the set of all {p\times q} real matrices with {p\geq q} whose ranks are strictly smaller than q. It is proved that {\Sigma_{pq}} is a minimal cone in {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.

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Simplex algebras and their representation

Glasgow Mathematical Journal ◽

10.1017/s0017089500001889 ◽

1973 ◽

Vol 14 (2) ◽

pp. 136-144

Author(s):

M. S. Vijayakumar

Keyword(s):

Left Ideal ◽

Maximal Ideal ◽

Banach Algebras ◽

Principal Component ◽

Nonempty Interior ◽

Regular Elements ◽

Maximal Left Ideal ◽

The Relationship ◽

Order Identity ◽

Real Matrices

This paper establishes a relationship (Theorem 4.1) between the approaches of A. C. Thompson [8, 9] and E. G. Effros [2] to the representation of simplex algebras, that is, real unital Banach algebras that are simplex spaces with the unit for order identity. It proves that the (nonempty) interior of the associated cone is contained in the principal component of the set of all regular elements of the algebra. It also conjectures that each maximal ideal (in the order sense—see below) of a simplex algebra contains a maximal left ideal of the algebra. This conjecture and other aspects of the relationship are illustrated by considering algebras of n × n real matrices.

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On a class of Jacobi-like procedures for diagonalising arbitrary real matrices

Numerische Mathematik ◽

10.1007/bf01399551 ◽

1979 ◽

Vol 33 (2) ◽

pp. 157-172 ◽

Cited By ~ 11

Author(s):

K. Veselić

Keyword(s):

Real Matrices

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Singularities of certain ladder determinantal varieties

Journal of Pure and Applied Algebra ◽

10.1016/0022-4049(94)00038-k ◽

1995 ◽

Vol 101 (1) ◽

pp. 59-75 ◽

Cited By ~ 6

Author(s):

Donna Glassbrenner ◽

Karen E. Smith

Keyword(s):

Determinantal Varieties

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Determinantal variety and normal embedding

Journal of Topology and Analysis ◽

10.1142/s1793525318500073 ◽

2017 ◽

Vol 10 (01) ◽

pp. 27-34 ◽

Cited By ~ 1

Author(s):

K. Katz ◽

M. Katz ◽

D. Kerner ◽

Y. Liokumovich

Keyword(s):

Metric Space ◽

Space Structure ◽

The Other ◽

Intrinsic Metric ◽

Lipschitz Equivalence ◽

Other Hand ◽

Determinantal Variety ◽

Conical Structure ◽

Positive Determinant

The space [Formula: see text] of matrices of positive determinant inherits an extrinsic metric space structure from [Formula: see text]. On the other hand, taking the infimum of the lengths of all paths connecting a pair of points in [Formula: see text] gives an intrinsic metric. We prove bi-Lipschitz equivalence between intrinsic and extrinsic metrics on [Formula: see text], exploiting the conical structure of the stratification of the space of [Formula: see text] matrices by rank.

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