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.
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© 2011 Springer Science+Business Media, LLC
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Yanai, H., Takeuchi, K., Takane, Y. (2011). Generalized Inverse Matrices. In: Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition. Statistics for Social and Behavioral Sciences. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9887-3_3
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DOI: https://doi.org/10.1007/978-1-4419-9887-3_3
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