Numerical Range of Moore–Penrose Inverse Matrices

Abstract

Let A be an n-by-n matrix. The numerical range of A is defined as W ( A ) = { x * A x : x ∈ C n , x * x = 1 } . The Moore–Penrose inverse A + of A is the unique matrix satisfying A A + A = A , A + A A + = A + , ( A A + ) * = A A + , and ( A + A ) * = A + A . This paper investigates the numerical range of the Moore–Penrose inverse A + of a matrix A, and examines the relation between the numerical ranges W ( A + ) and W ( A ) .

Keywords

• Moore–Penrose inverse
• numerical range
• weighted shift matrix