Generalized Inverse Matrices

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


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.


Linear Transformation Orthogonal Projector Simultaneous Equation Minimum Norm Generalize Inverse 
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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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