E if a1 does not exist then there are innitely many

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Unformatted text preview: exist, then there are infinitely many generalized inverses of A. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 13 / 27 An Algorithm for Finding a Generalized Inverse of a Matrix A 1 2 Find any r × r nonsingular submatrix of A where r = rank(A). Call this matrix W . Invert and transpose W , ie., compute (W −1 ) . 3 Replace each element of W in A with the corresponding element of (W −1 ) . 4 Replace all other elements in A with zeros. 5 Transpose the resulting matrix to obtain G, a generalized inverse for A. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 14 / 27 Invariance of PX = X(X X)− X to Choice of (X X)− If X X is nonsingular, then PX = X(X X)−1 X because the only generalized inverse of X X is (X X)−1 . If X X is singular, then PX = X(X X)− X and the choice of the generalized inverse (X X)− does not matter because PX = X(X X)− X will turn out to be the same matrix no matter which generalized inverse of X X is used. To see this, suppose (X X)− and (X X)− are any two generalized 1 2 inverses of X X. Then X(X X)− X = X(X X)− X X(X X)− X = X(X X)− X . 1 2 1 2 Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 15 / 27 An Example Orthogonal Projection Matrix Suppose y1 y2 1 1 = µ+ 1 , and we observe y = 2 6.1 2.3 . − = 1 1 1 1 1 1 [1 1] 1 1 = X (X X ) X 1 1 = − 1 1 = 1 2 = opyright c 2012 Dan Nettleton (Iowa State University) 1 1 − [1 1] [2]−1 [ 1 1 ] = 1 1 1 2 1 1 1 [1 1] 2 1 1 [1 1]= 1/2 1/2 1/2 1/2 1 1 . Statistics 511 16 / 27 An Example Orthogonal Projection Thus, the orthogonal projection of y = onto the column space of X = is PX y = 1/2 1/2 1/2 1/2 Copyright c 2012 Dan Nettleton (Iowa State University) 6.1 2.3 = 6.1 2.3 1 1 4.2 4.2 . Statistics 511 17 / 27 Why is PX called an orthogonal projection matrix? Suppose X = 1 2 and y = 2 3 4 . Xq Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 18 / 27 Why is PX called an orthogonal projection matrix? Suppose X = 1 2 and y = 2 3 4 . C (X...
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