ON THE RELATION BETWEEN MOORE’S AND PENROSE’S CONDITIONS

Moore (1920) defined the reciprocal of any matrix over the complex field by three conditions, but the beauty of the definition was not realized until Penrose (1955) defined the same inverse using four conditions. The reciprocal is now often called the MoorePenrose inverse, and has been widely used in various areas. This note comments on the definitions of Moore-Penrose inverse, and gives a new characterization for two types of weak Moore-Penrose inverses, which exposes an important relation between Moore’s and Penrose’s conditions. It also attempts to emphasize the merit of Moore’s definition, which has been overlooked mainly due to Moore’s unique notation. Two examples are given to demonstrate some combined applications of Moore’s and Penrose’s conditions, including a correction for an incorrect proof of Ben-Israel’s (1986) characterization for Moore’s conditions.


Introduction.
Denote by Ꮿ m×n the set of m × n finite matrices over the field of complex numbers, and by A * the conjugate transpose of a Ꮿ m×n -matrix A. Moore [4] defined the reciprocal Λ of a Ꮿ m×n -matrix K as a Ꮿ n×m -matrix such that (a) the columns of Λ are linear combinations of the conjugates of the rows of K, (b) the rows of Λ are linear combinations of the conjugates of the columns of K, (c) the matrix KΛK obtained by matricial composition of the matrices K, Λ, K is the original matrix K.In other words, Λ satisfies the following three conditions: (a) Λ = K * C for some C ∈ Ꮿ m×m , (b) Λ = DK * for some D ∈ Ꮿ n×n , (c) KΛK = K.Alternatively, if we denote by Col(K) the column space of K, Moore's conditions (a) and (b) can be written as (a) Col(Λ) ⊂ Col(K * ) and (b) Col(Λ * ) ⊂ Col(K).Furthermore, the existence and uniqueness of Λ were announced in [4], and proven in [5, pages 8, 202] after Moore's death, where the reciprocal was renamed as the general reciprocal.
However, the beauty of Moore's definition was not realized until Penrose [7] defined the generalized inverse A † of a matrix A ∈ Ꮿ m×n as the solution of the following four equations: (1) , and proved the existence and uniqueness of A † .Moore's publications were not cited in Penrose [7].Since then the A † has been widely used in various areas, especially in the study of approximate solutions (such as the least-squares solutions) of linear equations.
The equivalence between Moore's reciprocal and Penrose's generalized inverse was shown by Rado [10], and Penrose [8] noticed this too.The A † is now often called the Moore-Penrose inverse (M-P inverse) and sometimes is also called the pseudo-inverse.We denote by A † the M-P inverse of A.
Another equivalent definition of the M-P inverse, often used in algebraic textbooks, is as follows: if A is of rank r , {b 1 ,...,b r } is a basis of Col(A), and

.,b r ],
(hence A = BH for some H ∈ Ꮿ r ×n of rank r ), then (see Ben-Israel and Greville [2, page 23]).The existence of A † follows from this definition immediately; and if A is of full-rank, an algorithm for computing A † follows from the definition as well (see Noble [6, page 291]).But this definition does not describe the features of A † explicitly, while Moore's and Penrose's definitions do.
The weak M-P inverses (or weak generalized inverses) are also playing an important role in applications.We denote by A {i 1 ,...,i k } a weak M-P inverse of A satisfying Moore's or Penrose's conditions (i 1 ),...,(i k ) (hence A † ≡ A {a,b,c} ≡ A {1,2,3,4} and A {c} ≡ A {1} ), and by Ꮽ {i 1 ,...,i k } the set of all possible A {i 1 ,...,i k } 's (hence In addition to the comments on the definitions of M-P inverse, this note gives a new characterization for two types of weak M-P inverses, namely Ꮽ {a,c} = Ꮽ {1,2,4} and Ꮽ {b,c} = Ꮽ {1,2,3} , which exposes an important relation between Moore's and Penrose's conditions.It also attempts to emphasize the merit of Moore's definition, which has been overlooked mainly due to Moore's unique notation.Two examples are given to demonstrate some combined applications of Moore's and Penrose's conditions, including a correction for an incorrect proof of Ben-Israel's [1] characterization for Moore's conditions (a) and (b).

Characterization for two types of weak M-P inverses. Ben-Israel [1] examined
Penrose's A † in relation to Moore's A {a,b,c} , interpreted Moore's results in a more readable manner, and gave (as supported by an incorrect proof, which is corrected in Example 3.2.Pringle and Rayner [9, page 26] characterized weak M-P inverses A {1,2,3} and A {1,2,4} , using a result of Goldman and Zelen [3], and showed over the real field that for an m × n matrix A, where A denotes the transpose of A; and Plainly, (2.2) and (2.3) hold over the complex field.

Comment and examples.
Most research publications in the area of generalized inversion adopt Penrose's definition of four conditions, and overlook Moore's definition of three conditions.If Moore's unique notation was a reason for this phenomenon (see Ben-Israel [1], Pringle and Rayner [9, page 3]), it seems that we should pay more attention to Moore's definition now.
In fact, Moore's conditions (a) and (b) are quite intuitive and convenient to use.In some cases, combined applications of Moore's and Penrose's conditions can simplify approaches to a great extent.