Completely positive matrices.

*(English)*Zbl 1030.15022
River Edge, NJ: World Scientific. x, 206 p. (2003).

This monograph studies one small aspect of modern matrix theory in detail, namely the subject of completely positive matrices.

Its first chapter gives some of the necessary background in linear algebra and introduces the necessary tools of Perron-Frobenius theory, of the Schur complement, and of matrix cones and graphs.

Chapter two introduces and studies completely positive matrices. These are real positive semi-definite matrices \(A\) that allow a factorization of the form \(A = BB^T\) for an entry-wise non-negative matrix \(B\). While it is obvious how to construct such matrices, it is still open how one can determine whether a given matrix \(A\) is cp (short for completely positive). The results that are so far known about this problem are stated and proved in chapter two.

In chapter 3, a second open problem for cp matrices is presented, namely to find their minimal cp-rank. Here the cp-rank of a cp matrix \(A\) denotes the minimal number of columns needed for an entry-wise non-negative matrix \(B\) so that \(A = BB^T\). Again, the current state of our knowledge is described in a manner that is very suitable to study at the graduate level.

Each of the twenty sections of the book contains exercises; there is a list of references, one of the notations, and an index for the whole book. Overall, this appears to be a highly delightful book to read, study, and teach from.

Its first chapter gives some of the necessary background in linear algebra and introduces the necessary tools of Perron-Frobenius theory, of the Schur complement, and of matrix cones and graphs.

Chapter two introduces and studies completely positive matrices. These are real positive semi-definite matrices \(A\) that allow a factorization of the form \(A = BB^T\) for an entry-wise non-negative matrix \(B\). While it is obvious how to construct such matrices, it is still open how one can determine whether a given matrix \(A\) is cp (short for completely positive). The results that are so far known about this problem are stated and proved in chapter two.

In chapter 3, a second open problem for cp matrices is presented, namely to find their minimal cp-rank. Here the cp-rank of a cp matrix \(A\) denotes the minimal number of columns needed for an entry-wise non-negative matrix \(B\) so that \(A = BB^T\). Again, the current state of our knowledge is described in a manner that is very suitable to study at the graduate level.

Each of the twenty sections of the book contains exercises; there is a list of references, one of the notations, and an index for the whole book. Overall, this appears to be a highly delightful book to read, study, and teach from.

Reviewer: F.Uhlig (Auburn)

##### MSC:

15B48 | Positive matrices and their generalizations; cones of matrices |

15A03 | Vector spaces, linear dependence, rank, lineability |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

15A23 | Factorization of matrices |

15B51 | Stochastic matrices |