The positive semidefinite completion problem for two unspecified entries

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" The positive semidefinite completion problem for two unspecified entries "
M. O. N. a. Omran W. W. Barrett

Document Type	:	Latin Dissertation
Language of Document	:	English
Record Number	:	1113319
Doc. No	:	TLpq304343199
Main Entry	:	M. O. N. a. Omran
	:	W. W. Barrett
Title & Author	:	The positive semidefinite completion problem for two unspecified entries\ M. O. N. a. OmranW. W. Barrett
College	:	Brigham Young University
Date	:	1997
student score	:	1997
Degree	:	Ph.D.
Page No	:	53
Abstract	:	The positive semidefinite (PSD) completion problem is concerned with determining the set of PSD completions of a partial matrix. Let usdB(x,y)usd and usdA(x,y)usd be real partial PSD matrices of order usdn\ge4usd with x and y the two unspecified entries corresponding to the two adjacent (respectively, nonadjacent) missing edges of their graphs. We show that the possible completion regions for usdB(x,y)usd are either a single point, a line segment, or an ellipse. For usdA(x,y)usd we give a fairly complete description of the real PSD completion region R. We find necessary and sufficient conditions on the specified entries of usdA(x,y)usd so that there are 1, 2, 3 or 4 singular points on the boundary of R, and so that usdA(x,y)usd has rank 2 PSD completions. We show that rank 2 completions occur in one of three ways: either there is a unique PSD completion (R is a single point), or det usdA(x,y)usd factors (with the occurrence of singular points), or the PSD completion region R (not a single point) contains a unique rank 2 PSD completion which is a singular point.
Subject	:	Mathematics
	:	Pure sciences