Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem UWB Mathematics Society Pietro Paparella Division of Engineering & Mathematics School of Science, Technology, Engineering, & Mathematics 14 April 2016 B&N PFT NIEP SC/STFW Overview 1 Background & Notation 2 The Perron-Frobenius Theorem for Nonnegative Matrices 3 Nonnegative Inverse Eigenvalue Problem 4 Perron Spectracones & Spectratopes 5 Future Work P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. P. Paparella Perron Spectratopes E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. P. Paparella Perron Spectratopes A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. P. Paparella Perron Spectratopes B&N PFT NIEP SC/STFW N denotes the set of natural numbers. For n 2 N, denote by hni the set f1;:::; ng. R denotes the set of real numbers. C denotes the set of complex numbers. F denotes R or C. A matrix is a rectangular array of numbers arranged in rows and columns. If a matrix has m rows and n columns, then it is said to be an m-by-n matrix and its size is m-by-n. E.g., the matrix 1 2 0 A = −1 i π is a two-by-three matrix. A square matrix is any matrix with the same number of rows and columns (unless otherwise noted, all matrices are considered to be square). P. Paparella Perron Spectratopes the scalar λ 2 C is is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx. The vector x is called an eigenvector of A associated with λ. σ (A) denotes the spectrum of a A, i.e., σ (A) := fλ 2 C : det(A − λI ) = 0g = fλ1; : : : ; λng: ρ (A) denotes the spectral radius of A, i.e., ρ (A) := maxi2hni jλi j. aij denotes the (i; j)-entry of A (i.e., the entry in row i, column j). A> denotes the transpose of A, i.e., A> is the n-by-m matrix whose (i; j)-entry is aji . If A is an n-by-n matrix, then n For x 2 F , denote by Dx the diagonal matrix whose (i; i)-entry is xi . th n Denote by ei the i standard basis vector of R . Let e := [1 ··· 1]> 2 Rn. B&N PFT NIEP SC/STFW For any m-by-n matrix A: P. Paparella Perron Spectratopes the scalar λ 2 C is is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx. The vector x is called an eigenvector of A associated with λ. σ (A) denotes the spectrum of a A, i.e., σ (A) := fλ 2 C : det(A − λI ) = 0g = fλ1; : : : ; λng: ρ (A) denotes the spectral radius of A, i.e., ρ (A) := maxi2hni jλi j. A> denotes the transpose of A, i.e., A> is the n-by-m matrix whose (i; j)-entry is aji . If A is an n-by-n matrix, then n For x 2 F , denote by Dx the diagonal matrix whose (i; i)-entry is xi . th n Denote by ei the i standard basis vector of R . Let e := [1 ··· 1]> 2 Rn. B&N PFT NIEP SC/STFW For any m-by-n matrix A: aij denotes the (i; j)-entry of A (i.e., the entry in row i, column j). P. Paparella Perron Spectratopes the scalar λ 2 C is is called an eigenvalue of A if there is a nonzero vector x such that Ax = λx. The vector x is called an eigenvector of A associated with λ. σ (A) denotes the spectrum of a A, i.e., σ (A) := fλ 2 C : det(A − λI ) = 0g = fλ1; : : : ; λng: ρ (A) denotes the spectral radius of A, i.e., ρ (A) := maxi2hni jλi j.