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Generalized Inverse Matrices

Chapter
Part of the Statistics for Social and Behavioral Sciences book series (SSBS)

Abstract

Let A be a square matrix of order n. If it is nonsingular, then Ker(A) = {0} and, as mentioned earlier, the solution vector x in the equation y = Ax is determined uniquely as x = A -1 y. Here, A -1 is called the inverse (matrix) of A defining the inverse transformation from y ∈ En to x ∈ Em, whereas the matrix A represents a transformation from x to y. When A is n by m, Ax = y has a solution if and only if y ∈ Sp(A). Even then, if Ker(A) ≠ {A}, there are many solutions to the equation Ax = A due to the existence of x 0 (≠ 0) such that Ax 0 = 0, so that A(x+x 0) = y. If y ∉ Sp(A), there is no solution vector to the equation Ax = y.

Keywords

Linear Transformation Orthogonal Projector Simultaneous Equation Minimum Norm Generalize Inverse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of StatisticsSt. Luke’s College of NursingTokyoJapan
  2. 2.KamakurashiJapan
  3. 3.Department of PsychologyMcGill UniversityMontrealCanada

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