Algebraically positive matrices

Sign patterns that allow algebraic positivity

“…Theorem 1.1 ( [5]). A real square matrix is algebraically positive if and only if it has a simple real eigenvalue and corresponding left and right positive eigenvectors.…”

“…Kirkland, Qiao, and Zhan [5] characterized symmetric tridiagonal sign pattern matrices that allow algebraic positivity. Later, Das and Bandopadhyay [2] generalized this result to find all tree sign pattern matrices allowing algebraic positivity, and Abagat and Pelejo [1] described all 3-by-3 sign pattern matrices that allow algebraic positivity.…”

On Sign Pattern Matrices that Allow or Require Algebraic Positivity

“…Some natural generalizations of the concept of matrix positivity have also been studied extensively such as matrix nonnegativity, matrix primitivity and eventual positivity. In [3], Kirkland, Qiao and Zhan presented another generalization called algebraic positivity. A matrix A ∈ M n is algebraically positive if p(A) > 0 for some real polynomial p. Using the Cayley-Hamilton theorem, it can be deduced that a matrix A ∈ M n , where n ≥ 2, is algebraically positive if and only if there are real numbers k 1 , .…”

“…The authors of [3] used the Perron-Frobenius theorem to prove the following characterization of algebraically positive matrices. Theorem 1.1 (Kirkland, Qiao, and Zhan, 2016).…”

“…The authors of [3] used the Perron-Frobenius theorem to prove the following characterization of algebraically positive matrices. Theorem 1.1 (Kirkland, Qiao, and Zhan, 2016). A real matrix is algebraically positive if and only if it has a simple real eigenvalue and corresponding left and right positive eigenvectors.…”