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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