On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.

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Preservers of Completely Positive Matrix Rank for Inclines

Bulletin of the Malaysian Mathematical Sciences Society ◽

10.1007/s40840-017-0490-z ◽

2017 ◽

Vol 42 (2) ◽

pp. 437-447 ◽

Cited By ~ 1

Author(s):

LeRoy B. Beasley ◽

Preeti Mohindru ◽

Rajesh Pereira

Keyword(s):

Positive Matrix ◽

Completely Positive ◽

Matrix Rank ◽

Completely Positive Matrix

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Bounding the CP-rank by graph parameters

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3009 ◽

2015 ◽

Vol 28 ◽

Author(s):

Naomi Shaked-Monderer

Keyword(s):

Lower Bound ◽

Connected Graph ◽

Maximum Size ◽

Covering Number ◽

Positive Matrix ◽

Completely Positive ◽

Completely Positive Matrix ◽

Graph Parameters

The cp-rank of a graph G, cpr(G), is the maximum cp-rank of a completely positive matrix with graph G. One obvious lower bound on cpr(G) is the (edge-) clique covering number, cc(G), i.e., the minimal number of cliques needed to cover all of Gâs edges. It is shown here that for a connected graph G, cpr(G) = cc(G) if and only if G is triangle free and not a tree. Another lower bound for cpr(G) is tf(G), the maximum size of a triangle free subgraph of G. We consider the question of when does the equality cpr(G) = tf(G) hold.

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Preservers of completely positive matrix rank

Linear and Multilinear Algebra ◽

10.1080/03081087.2015.1082960 ◽

2015 ◽

Vol 64 (7) ◽

pp. 1258-1265

Author(s):

Seok-Zun Song ◽

LeRoy B. Beasley ◽

Preeti Mohindru ◽

Rajesh Pereira

Keyword(s):

Positive Matrix ◽

Completely Positive ◽

Matrix Rank ◽

Completely Positive Matrix

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Note on Guttman Unidimensional Scaling

Psychological Reports ◽

10.2466/pr0.1966.19.3.767 ◽

1966 ◽

Vol 19 (3) ◽

pp. 767-770

Author(s):

Arthur D. Kirsch

Keyword(s):

Positive Matrix ◽

Completely Positive ◽

Unidimensional Scaling ◽

Completely Positive Matrix

This paper reviews the concept of Guttman cumulative or unidimensional scaling and presents an added requirement that such scales must have a completely positive matrix of item intercorrelation. 1

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On the number of CP factorizations of a completely positive matrix

Linear and Multilinear Algebra ◽

10.1080/03081087.2020.1856766 ◽

2021 ◽

pp. 1-18

Author(s):

Naomi Shaked-Monderer

Keyword(s):

Positive Matrix ◽

Completely Positive ◽

Completely Positive Matrix

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Distance Matrix of a Class of Completely Positive Graphs: Determinant and Inverse

Special Matrices ◽

10.1515/spma-2020-0109 ◽

2020 ◽

Vol 8 (1) ◽

pp. 160-171

Author(s):

Joyentanuj Das ◽

Sachindranath Jayaraman ◽

Sumit Mohanty

Keyword(s):

Symmetric Matrix ◽

Nonnegative Matrix ◽

Distance Matrix ◽

Laplacian Matrix ◽

Positive Semidefinite ◽

Positive Matrix ◽

Completely Positive ◽

Rank One ◽

Principal Submatrices ◽

Existing Result

AbstractA real symmetric matrix A is said to be completely positive if it can be written as BBt for some (not necessarily square) nonnegative matrix B. A simple graph G is called a completely positive graph if every matrix realization of G that is both nonnegative and positive semidefinite is a completely positive matrix. Our aim in this manuscript is to compute the determinant and inverse (when it exists) of the distance matrix of a class of completely positive graphs. We compute a matrix 𝒭 such that the inverse of the distance matrix of a class of completely positive graphs is expressed a linear combination of the Laplacian matrix, a rank one matrix of all ones and 𝒭. This expression is similar to the existing result for trees. We also bring out interesting spectral properties of some of the principal submatrices of 𝒭.

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