The Moore–Penrose inverse: a hundred years on a frontline of physics research

Abstract

The Moore–Penrose inverse celebrated its 100th birthday in 2020, as the notion standing behind the term was first defined by Eliakim Hastings Moore in 1920 (Bull Am Math Soc 26:394–395, 1920). Its rediscovery by Sir Roger Penrose in 1955 (Proc Camb Philos Soc 51:406–413, 1955) can be considered as a caesura, after which the inverse attracted the attention it deserves and has henceforth been exploited in various research branches of applied origin. The paper contemplates the role, which the Moore–Penrose inverse plays in research within physics and related areas at present. An overview of the up-to-date literature leads to the conclusion that the inverse “grows” along with the development of physics and permanently (maybe even more demonstrably now than ever before) serves as a powerful and versatile tool to cope with the current research problems.

Application of Generalized Inverses in the Minimum-Energy Perfect Control Theory

“…The matrix inverse and generalized inverse formulas have been utilized in a number of scientific and engineering fields, going far beyond the control-oriented applications, to mention physics [38], medicine [45], or economics [46]. Moreover, at the start of this section, it should be recalled and strongly highlighted that the MP generalized inverses have without a doubt been treated as the optimal ones, since they minimize the Euclidean norm in every general case [31], [32], [38], [39]. However, based on the new analytical investigation conducted in the previous section, we can now challenge the optimal peculiarity of the minimum-norm MP pseudoinverse.…”

“…A considerable potential of a practical use of the presented results is also corroborated by the representative numerical examples. The original observation covering the d-PC law breaks down the well-established IMC paradigm associated with the "pseudo-optimal" MP inverse, as a natural extension of the usual inverse [37], [38], and opens a new chapter in the control and systems theory canons.…”

Aplikasi Dekomposisi Rank Pada Pembentukan Invers Moore-Penrose Matriks Atas Aljabar Max-Plus Tersimetri

“…Untuk sistem persamaan linier 𝐴𝑥 = 𝑏 dengan 𝐴 + adalah invers Moore-Penrose dari 𝐴, maka solusi pendekatan dari sistem persamaan linier tersebut adalah 𝑥 = 𝐴 + 𝑏 (Campbell & Meyer, 2009). Beberapa penelitian yang berkaitan dengan pengembangan invers Moore-Penrose pada aljabar linier sudah dilakukan, antara lain review invers Moore-Penrose sebagai invers semu suatu matriks (Barata & Hussein, 2012), invers Moore-Penrose pada penelitian fisika (Baksalary & Trenkler, 2021) dan masalah pertubasi invers Moore-Penrose suatu matriks (Xu, 2019).…”

Low-cost fiber Bragg grating interrogation with a femtosecond laser written scattering chip

“…Whilst it may seem trivial to solve this equation, the inverse of a nonsquare matrix is non-existent (which is true for almost all the cases, since total number of single-speckle pixels is much greater than the number of captured speckles). The solution to this computation problem is to use the Moore-Penrose pseudo-inverse of the matrix C, which can be calculated from the singular value decomposition: [16][17][18] U and V are unitary matrices, Σ is a diagonal matrix with the singular values and T denotes the matrix transpose. Matrix Σ −1 is obtained by reciprocating the main diagonal of the Σ matrix (singular values) and then transposing the resulted matrix.…”