E if a1 does not exist then there are innitely many

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online