Question 2 (Cones of Matrices, of all zeros, as appropriate by context. Let S, := {A € R"*": A = AT} be the set of all symmetric real n x n matrices. Let S := {AE S,: A 2 0} be the set of nonnegative symmetric matrices, meaning that every such matrix has cach entry in it 2 0. We defined the cone of symmetric positive semidefinite matrices in lectures and we denote this by PSD. Now consider the following sets of matrices, Throughout, let 0 denote either a vector or matrix DNN, = PSD,,ns COP, = {A € Sn: v" Av 2 0, Ve 2 0} CPn = {A € S,: A = BB' for some B 2 0} (Doubly nonnegative matrices) (Copositive matrices) (Completely positive matrices) Note that in the definition of CP,n, the matrix B is of dimension n x k for some integer k. 1. What is the difference between a psd matrix and a copositive matrix ?
Question 2 (Cones of Matrices, of all zeros, as appropriate by context. Let S, := {A € R"*": A = AT} be the set of all symmetric real n x n matrices. Let S := {AE S,: A 2 0} be the set of nonnegative symmetric matrices, meaning that every such matrix has cach entry in it 2 0. We defined the cone of symmetric positive semidefinite matrices in lectures and we denote this by PSD. Now consider the following sets of matrices, Throughout, let 0 denote either a vector or matrix DNN, = PSD,,ns COP, = {A € Sn: v" Av 2 0, Ve 2 0} CPn = {A € S,: A = BB' for some B 2 0} (Doubly nonnegative matrices) (Copositive matrices) (Completely positive matrices) Note that in the definition of CP,n, the matrix B is of dimension n x k for some integer k. 1. What is the difference between a psd matrix and a copositive matrix ?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Question 2 (Cones of Matrices,
of all zeros, as appropriate by context.
Let S, := {A € R"*": A = AT} be the set of all symmetric real n x n matrices. Let S := {A E
Sn: A 2 0} be the set of nonnegative symmetric matrices, meaning that every such matrix has each
Throughout, let 0 denote either a vector or matrix
entry in it 2 0. We defined the cone of symmetric positive semidefinite matrices in lectures and we
denote this by PSD„. Now consider the following sets of matrices,
:= PSD, n s
COP. Α ε S,: υΤΑυ > 0, Vu 2 0
CPn := {A € Sn: A = BB" for some B 2 0}
DNN.
(Doubly nonnegative matrices)
(Copositive matrices)
(Completely positive matrices)
Note that in the definition of CPn, the matrix B is of dimension n x k for some integer k.
1. What is the difference between a psd matrix and a copositive matrix ?
'An extreme point is a point that cannot be written as a convex combination of two other points.
2This was covered in week 2 in Quadratic. pdf.
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