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 ) .
Highlights
Let A ∈ Mm,n, the m × n complex matrices, the Moore–Penrose inverse A+ is the unique matrix that satisfies the following properties [1,2]: AA+ A = A, A+ AA+ = A+, ( AA+ )∗ = AA+, and ( A+ A)∗ = A+ A.Consider the system of linear equations: Ax = b, b ∈ Cn .Moore and Penrose showed that A+ b is a vector x such that k x k2 is minimized among all vectors x for which k Ax − bk2 is minimal
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)
We investigate the numerical ranges of the Moore–Penrose inverses, Mathematics 2020, 8, 830; doi:10.3390/math8050830
Summary
Let A ∈ Mm,n , the m × n complex matrices, the Moore–Penrose inverse A+ is the unique matrix that satisfies the following properties [1,2]: AA+ A = A, A+ AA+ = A+ , ( AA+ )∗ = AA+ , and ( A+ A)∗ = A+ A. Let Mn be the set of n × n complex matrices. The numerical range of A ∈ Mn is defined as. There are several fundamental facts about the numerical ranges of square matrices:. We investigate the numerical ranges of the Moore–Penrose inverses, Mathematics 2020, 8, 830; doi:10.3390/math8050830 www.mdpi.com/journal/mathematics. The following facts list a number of useful properties concerning the Moore–Penrose inverse. Assume A = UΣV ∗ ∈ Mm,n is a singular value decomposition of A, A+ = VΣ+ U ∗.
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