Semimonotone Matrices Megan Wendler May 27, 2018 Megan Wendler Semimonotone Matrices May 27, 2018 1 / 37 Table of contents 1 Introduction The definition of semimonotone & an example Some observations and previous results Questions 2 Some Results What kinds of matrices are semimonotone? Properties of semimonotone matrices 3 Conjectures 4 Future Directions Megan Wendler Semimonotone Matrices May 27, 2018 2 / 37 Outline 1 Introduction The definition of semimonotone & an example Some observations and previous results Questions 2 Some Results What kinds of matrices are semimonotone? Properties of semimonotone matrices 3 Conjectures 4 Future Directions Megan Wendler Semimonotone Matrices May 27, 2018 3 / 37 Definition A matrix A 2 Mn(R) is called strictly semimonotone if (Ax)k ≥ 0 is replaced with (Ax)k > 0 in the above definition. The Definition of Semimonotone & Strictly Semimonotone Definition A matrix A 2 Mn(R) is semimonotone if n 0 6= x ≥ 0 where x 2 R ) xk > 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 2 Mn(R) is semimonotone if n 0 6= x ≥ 0 where x 2 R ) xk > 0 and (Ax)k ≥ 0 for some k Definition A matrix A 2 Mn(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 x1 2 Let x = 2 R where 0 6= x ≥ 0. x2 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 0; a contradiction. Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Example of a Semimonotone Matrix Example 2 −1 Let A = . −2 3 Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 0; a contradiction. Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Example of a Semimonotone Matrix Example 2 −1 x1 2 Let A = . Let x = 2 R where 0 6= x ≥ 0. −2 3 x2 Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37 Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 0; a contradiction. Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Example of a Semimonotone Matrix Example 2 −1 x1 2 Let A = . Let x = 2 R where 0 6= x ≥ 0. −2 3 x2 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37 Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 0; a contradiction. Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Example of a Semimonotone Matrix Example 2 −1 x1 2 Let A = . Let x = 2 R where 0 6= x ≥ 0. −2 3 x2 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 0; a contradiction. Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Example of a Semimonotone Matrix Example 2 −1 x1 2 Let A = . Let x = 2 R where 0 6= x ≥ 0. −2 3 x2 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37 Thus, we have shown that A is semimonotone. In fact, A is strictly semimonotone. Example of a Semimonotone Matrix Example 2 −1 x1 2 Let A = . Let x = 2 R where 0 6= x ≥ 0. −2 3 x2 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 0; a contradiction. Megan Wendler Semimonotone Matrices May 27, 2018 5 / 37 Example of a Semimonotone Matrix Example 2 −1 x1 2 Let A = . Let x = 2 R where 0 6= x ≥ 0. −2 3 x2 Clearly, if x1 = 0, then we must have that x2 > 0 and we get that (Ax)2 = 3x2 > 0. Similarly, if x2 = 0, then we must have that x1 > 0 and we get that (Ax)1 = 2x1 > 0. Now suppose x1; x2 > 0. In this case, 2x − x Ax = 1 2 −2x1 + 3x2 Suppose Ax < 0. Then x2 > 2x1. Thus, we must have −2x1 + 3x2 > −2x1 + 3(2x1) = 4x1 > 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 1 Introduction The definition of semimonotone & an example Some observations and previous results Questions 2 Some Results What kinds of matrices are semimonotone? Properties of semimonotone matrices 3 Conjectures 4 Future Directions Megan Wendler Semimonotone Matrices May 27, 2018 6 / 37 Every principal submatrix A(α; α) must be semimonotone, where α ⊆ f1; 2;:::; ng. (This can be shown by taking any x where x[α] > 0 and all the other entries are zero.) Proposition A matrix A 2 Mn(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. A few simple observations about a semimonotone matrix Suppose A 2 Mn(R) is semimonotone. By letting x = ek , we obtain that akk ≥ 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 Proposition A matrix A 2 Mn(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. A few simple observations about a semimonotone matrix Suppose A 2 Mn(R) is semimonotone. By letting x = ek , we obtain that akk ≥ 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 ⊆ f1; 2;:::; ng. (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 2 Mn(R) is semimonotone. By letting x = ek , we obtain that akk ≥ 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 ⊆ f1; 2;:::; ng. (This can be shown by taking any x where x[α] > 0 and all the other entries are zero.) Proposition A matrix A 2 Mn(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 Definition T A matrix A 2 Mn(R) is copositive if x Ax ≥ 0 for all x ≥ 0. Definition A matrix A 2 Mn(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 2 Mn(R) is said to be an S0 matrix if there exists a 0 6= x ≥ 0 such that Ax ≥ 0. Previous Results Before we discuss previous results, we need to first recall some definitions. Definition A matrix A 2 Mn(R) is a P-matrix (P0-matrix) if all its principal minors are positive (nonnegative). Megan Wendler Semimonotone Matrices May 27, 2018 8 / 37 Definition A matrix A 2 Mn(R) is semipositive if there exists an x ≥ 0 such that Ax > 0.