How to get the pseudo left inverse of a singular squarem atrix?

I'm rusty, but not *that* rusty here, I hope.

If W (=Z*Z' in your case) is singular, it can not
have >inverse, which by
definition also mean that nothing multiply by it will
produce the identity
matrix (for otherwise it would have an inverse and
thus nonsingular).

The definition of a generalized inverse is something
like:  If A is a
non-null matrix, and G satisfy AGA = A, then G is
called a generalized
inverse of A.  This is not unique, but a unique one
that satisfy some
additional properties is the Moore-Penrose inverse. 
I >don't know if this is
what ginv() in MASS returns, as I have not used it
before.
Andy

The inverse of a Matrix A is defined as a Matrix B
such that B*A=A*B=I and not just B*A=I. But there are
matrices B for singular matrices A such that B*A=I but
A*B != I, therefore there exist "left-inverses" (or
"right-inverses") for non-invertable matrices.

Best Regards

__________________________________
The documentation for "ginv" in MASS says it "Calculates the 
Moore-Penrose generalized inverse of a matrix 'X'."  The theory says 
that for each m x n matrix A, there is a unique n x m matrix G 
satisfying AGA = A and GAG = G.  
(http://mathworld.wolfram.com/Moore-PenroseMatrixInverse.html). 

      Consider the following simple example:
 A <- array(c(1,1,0,0), dim=c(2,2))
[2,]  0.0  0.0
 A
[,1] [,2]
[1,]    1    0
[2,]    1    0
 ginv(A)
[,1] [,2]
[1,]  0.5  0.5
[2,]  0.0  0.0
 ginv(A)%*%A
[,1] [,2]
[1,]    1    0
[2,]    0    0
 A%*%ginv(A)
[,1] [,2]
[1,]  0.5  0.5
[2,]  0.5  0.5
 A%*%ginv(A)%*%A
[,1] [,2]
[1,]    1    0
[2,]    1    0
 ginv(A)%*%A%*%ginv(A)
[,1] [,2]
[1,]  0.5  0.5
[2,]  0.0  0.0

      hope this helps.  spencer graves

I'm rusty, but not *that* rusty here, I hope.

If W (=Z*Z' in your case) is singular, it can not

have >inverse, which by

definition also mean that nothing multiply by it will
produce the identity
matrix (for otherwise it would have an inverse and
thus nonsingular).

The definition of a generalized inverse is something
like:  If A is a
non-null matrix, and G satisfy AGA = A, then G is
called a generalized
inverse of A.  This is not unique, but a unique one
that satisfy some
additional properties is the Moore-Penrose inverse. 

I >don't know if this is

what ginv() in MASS returns, as I have not used it
before.

Andy

The inverse of a Matrix A is defined as a Matrix B
such that B*A=A*B=I and not just B*A=I. But there are
matrices B for singular matrices A such that B*A=I but
A*B != I, therefore there exist "left-inverses" (or
"right-inverses") for non-invertable matrices.

Best Regards

__________________________________

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