Semimonotone Matrices Megan Wendler May 27, 2018 Megan Wendler - PowerPoint PPT Presentation

Semimonotone Matrices Megan Wendler May 27, 2018 Megan Wendler Semimonotone Matrices May 27, 2018 1 / 37

Table of contents Introduction 1 The definition of semimonotone & an example Some observations and previous results Questions Some Results 2 What kinds of matrices are semimonotone? Properties of semimonotone matrices Conjectures 3 Future Directions 4 Megan Wendler Semimonotone Matrices May 27, 2018 2 / 37

Outline Introduction 1 The definition of semimonotone & an example Some observations and previous results Questions Some Results 2 What kinds of matrices are semimonotone? Properties of semimonotone matrices Conjectures 3 Future Directions 4 Megan Wendler Semimonotone Matrices May 27, 2018 3 / 37

The Definition of Semimonotone & Strictly Semimonotone Definition A matrix A ∈ M n ( R ) is semimonotone if 0 � = x ≥ 0 where x ∈ R n ⇒ x k > 0 and ( Ax ) k ≥ 0 for some k Megan Wendler Semimonotone Matrices May 27, 2018 4 / 37

The Definition of Semimonotone & Strictly Semimonotone Definition A matrix A ∈ M n ( R ) is semimonotone if 0 � = x ≥ 0 where x ∈ R n ⇒ x k > 0 and ( Ax ) k ≥ 0 for some k Definition A matrix A ∈ M n ( R ) is called strictly semimonotone if ( Ax ) k ≥ 0 is replaced with ( Ax ) k > 0 in the above definition. Megan Wendler Semimonotone Matrices May 27, 2018 4 / 37

Example of a Semimonotone Matrix Example � 2 � − 1 Let A = . − 2 3 Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Example of a Semimonotone Matrix Example � 2 � � x 1 � − 1 ∈ R 2 where 0 � = x ≥ 0. Let A = . Let x = − 2 3 x 2 Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Example of a Semimonotone Matrix Example � 2 � � x 1 � − 1 ∈ R 2 where 0 � = x ≥ 0. Let A = . Let x = − 2 3 x 2 Clearly, if x 1 = 0, then we must have that x 2 > 0 and we get that ( Ax ) 2 = 3 x 2 > 0. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Example of a Semimonotone Matrix Example � 2 � � x 1 � − 1 ∈ R 2 where 0 � = x ≥ 0. Let A = . Let x = − 2 3 x 2 Clearly, if x 1 = 0, then we must have that x 2 > 0 and we get that ( Ax ) 2 = 3 x 2 > 0. Similarly, if x 2 = 0, then we must have that x 1 > 0 and we get that ( Ax ) 1 = 2 x 1 > 0. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Example of a Semimonotone Matrix Example � 2 � � x 1 � − 1 ∈ R 2 where 0 � = x ≥ 0. Let A = . Let x = − 2 3 x 2 Clearly, if x 1 = 0, then we must have that x 2 > 0 and we get that ( Ax ) 2 = 3 x 2 > 0. Similarly, if x 2 = 0, then we must have that x 1 > 0 and we get that ( Ax ) 1 = 2 x 1 > 0. Now suppose x 1 , x 2 > 0. In this case, � 2 x 1 − x 2 � Ax = − 2 x 1 + 3 x 2 Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Example of a Semimonotone Matrix Example � 2 � � x 1 � − 1 ∈ R 2 where 0 � = x ≥ 0. Let A = . Let x = − 2 3 x 2 Clearly, if x 1 = 0, then we must have that x 2 > 0 and we get that ( Ax ) 2 = 3 x 2 > 0. Similarly, if x 2 = 0, then we must have that x 1 > 0 and we get that ( Ax ) 1 = 2 x 1 > 0. Now suppose x 1 , x 2 > 0. In this case, � 2 x 1 − x 2 � Ax = − 2 x 1 + 3 x 2 Suppose Ax < 0. Then x 2 > 2 x 1 . Thus, we must have − 2 x 1 + 3 x 2 > − 2 x 1 + 3(2 x 1 ) = 4 x 1 > 0 , a contradiction. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Example of a Semimonotone Matrix Example � 2 � � x 1 � − 1 ∈ R 2 where 0 � = x ≥ 0. Let A = . Let x = − 2 3 x 2 Clearly, if x 1 = 0, then we must have that x 2 > 0 and we get that ( Ax ) 2 = 3 x 2 > 0. Similarly, if x 2 = 0, then we must have that x 1 > 0 and we get that ( Ax ) 1 = 2 x 1 > 0. Now suppose x 1 , x 2 > 0. In this case, � 2 x 1 − x 2 � Ax = − 2 x 1 + 3 x 2 Suppose Ax < 0. Then x 2 > 2 x 1 . Thus, we must have − 2 x 1 + 3 x 2 > − 2 x 1 + 3(2 x 1 ) = 4 x 1 > 0 , a contradiction. Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37

Outline Introduction 1 The definition of semimonotone & an example Some observations and previous results Questions Some Results 2 What kinds of matrices are semimonotone? Properties of semimonotone matrices Conjectures 3 Future Directions 4 Megan Wendler Semimonotone Matrices May 27, 2018 6 / 37

A few simple observations about a semimonotone matrix Suppose A ∈ M n ( R ) is semimonotone. By letting x = e k , we obtain that a kk ≥ 0, for each k = 1 , 2 , . . . , n . This means that the diagonal entries of A must be nonnegative. Megan Wendler Semimonotone Matrices May 27, 2018 7 / 37

A few simple observations about a semimonotone matrix Suppose A ∈ M n ( R ) is semimonotone. By letting x = e k , we obtain that a kk ≥ 0, for each k = 1 , 2 , . . . , n . This means that the diagonal entries of A must be nonnegative. Every principal submatrix A ( α, α ) must be semimonotone, where α ⊆ { 1 , 2 , . . . , n } . (This can be shown by taking any x where x [ α ] > 0 and all the other entries are zero.) Megan Wendler Semimonotone Matrices May 27, 2018 7 / 37

A few simple observations about a semimonotone matrix Suppose A ∈ M n ( R ) is semimonotone. By letting x = e k , we obtain that a kk ≥ 0, for each k = 1 , 2 , . . . , n . This means that the diagonal entries of A must be nonnegative. Every principal submatrix A ( α, α ) must be semimonotone, where α ⊆ { 1 , 2 , . . . , n } . (This can be shown by taking any x where x [ α ] > 0 and all the other entries are zero.) Proposition A matrix A ∈ M n ( R ) is semimonotone if and only if 1 Every proper principal submatrix of A is semimonotone, and 2 For every x > 0 , ( Ax ) k ≥ 0 for some k. Megan Wendler Semimonotone Matrices May 27, 2018 7 / 37

Previous Results Before we discuss previous results, we need to first recall some definitions. Definition A matrix A ∈ M n ( R ) is a P -matrix ( P 0 -matrix) if all its principal minors are positive (nonnegative). Megan Wendler Semimonotone Matrices May 27, 2018 8 / 37

Previous Results Before we discuss previous results, we need to first recall some definitions. Definition A matrix A ∈ M n ( R ) is a P -matrix ( P 0 -matrix) if all its principal minors are positive (nonnegative). Definition A matrix A ∈ M n ( R ) is copositive if x T Ax ≥ 0 for all x ≥ 0. Megan Wendler Semimonotone Matrices May 27, 2018 8 / 37

Previous Results Before we discuss previous results, we need to first recall some definitions. Definition A matrix A ∈ M n ( R ) is a P -matrix ( P 0 -matrix) if all its principal minors are positive (nonnegative). Definition A matrix A ∈ M n ( R ) is copositive if x T Ax ≥ 0 for all x ≥ 0. Definition A matrix A ∈ M n ( R ) is semipositive if there exists an x ≥ 0 such that Ax > 0. By continuity of a matrix as a linear map, this is equivalent to saying that there exists an x > 0 such that Ax > 0. The class of semipositive matrices is denoted S . Megan Wendler Semimonotone Matrices May 27, 2018 8 / 37

Previous Results Before we discuss previous results, we need to first recall some definitions. Definition A matrix A ∈ M n ( R ) is a P -matrix ( P 0 -matrix) if all its principal minors are positive (nonnegative). Definition A matrix A ∈ M n ( R ) is copositive if x T Ax ≥ 0 for all x ≥ 0. Definition A matrix A ∈ M n ( R ) is semipositive if there exists an x ≥ 0 such that Ax > 0. By continuity of a matrix as a linear map, this is equivalent to saying that there exists an x > 0 such that Ax > 0. The class of semipositive matrices is denoted S . Definition A matrix A ∈ M n ( R ) is said to be an S 0 matrix if there exists a 0 � = x ≥ 0 such that Ax ≥ 0. Megan Wendler Semimonotone Matrices May 27, 2018 8 / 37

Previous Results Semimonotone matrices have been studied a little in the past. Below are some useful results obtained by Cottle, Pang, and Stone [1]: Megan Wendler Semimonotone Matrices May 27, 2018 9 / 37

Previous Results Semimonotone matrices have been studied a little in the past. Below are some useful results obtained by Cottle, Pang, and Stone [1]: 1 Every nonnegative matrix is semimonotone. Megan Wendler Semimonotone Matrices May 27, 2018 9 / 37

Previous Results Semimonotone matrices have been studied a little in the past. Below are some useful results obtained by Cottle, Pang, and Stone [1]: 1 Every nonnegative matrix is semimonotone. 2 Every P 0 -matrix is semimonotone. Every P -matrix is strictly semimonotone. Megan Wendler Semimonotone Matrices May 27, 2018 9 / 37

Previous Results Semimonotone matrices have been studied a little in the past. Below are some useful results obtained by Cottle, Pang, and Stone [1]: 1 Every nonnegative matrix is semimonotone. 2 Every P 0 -matrix is semimonotone. Every P -matrix is strictly semimonotone. 3 All copositive matrices are semimonotone. Megan Wendler Semimonotone Matrices May 27, 2018 9 / 37