Matrix Completions Problems for Patterns

Oct. 05
This page is intended to provide current information about the state of knowledge for the matrix completion problem for patterns for various classes of matrices.  Please e-mail any questions, corrections, suggestions or contributions to



Current State of Knowledge for Various Classes of Matrices
Positive Definite Matrices Positive Semi-Definite(PSD) Matrices

Eudclidan Distance Matrices

Completely Positive (CP) Matrices Doubly Nonnegative (DN) Matrices
Strictly Copositive

M-Matrices M0-Matrices
Symmetric M-matrices Symmetric M0-matrices
Inverse M-Matrices Singular Inverse M-matrices  (TCIM-matrices)
Symmetric Inverse M-matrices

P-Matrices P0-Matrices P0,1-Matrices
Positive (Nonnegative) P-matrices Nonnegative P0 (nnP0)-matrices
Sign symmetric P (ssP) Matrices Sign-symmetric P0 (ssP0) Matrices Sign Symmetric P0,1-Matrices
Weakly sign symmetric P (wssP) Matrices Weakly sign symmetric P0 (wssP0) Matrices Weakly Sign Symmetric P0,1-Matrices

If the class you want is not linked, e-mail for information.
The one-to-two page PDF files for individual classes contain the following information:

  • Status: summary of current state of knowledge (e.g., done, little progress, etc.)
  • Definition of the class of matrices.
  • Definition of a partial matrix in the class
  • Results: more detailed description of what is known, including citations to references.
  • Examples of digraph or graph diagrams, having completion and not having completion.  On the diagram, a vertex v with (v,v) included in the pattern (a specified vertex) is indicated by a solid black dot and a vertex v with (v,v) omitted from the pattern (an unspecified vertex) will be indicated by a hollow circle.  For a symmetric class, only positionally symmetric patterns are relevant, and diagrams are graph diagrams; otherwise they are digraph diagrams.  In a digraph, when both arcs (v,w) and (w,v) are present in a digraph, the arrows can be omitted, and this is represented by a double line.  Thus for a positionally symmetric pattern for a nonsymmetric class a digraph diagram with double edges is used.
  • References
  • Links to Other Sites

    Iowa State University 
    Combinatorial Matrix Group
    Leslie Hogben's Research Shaun Fallat's Research
    Topics in Linear Algebra Conference International Linear Algebra Society (ILAS) Electronic Journal of Linear Algebra (ELA)

    This page is designed and maintained by Leslie Hogben with contributions from Luz DeAlba, Amy Wangsness, and Shaun Fallat.
    Please e-mail any corrections, suggestions or contributions to