While working with linear models,I came across the situation where the proof of some results might be easier if we apply that transpose of g inverse is g inverse of transpose of a matrix.In R this codes
gave me
.But we know the g inverse of a matrix is not unique. Also we note the condition of g inverse is
\(AGA=A\)
\(A^tG^tA^t=A^t\)
So Shall we restrict ourselves by saying that if G is the G inverse of A then\( G^t\) is one of the g inverse of \(A^t \)? or more generally we can say that
\((A^-)^t=(A^t)^-\)
just like ordinary inverses?Also any link for studying this will be helpful.
Code:
library(MASS)
x1=c(1,0,1,1)
x2=c(3,6,5,2)
x3=c(2,4,6,1)
x4=c(3,5,2,1)
m1=cbind(x1,x2,x3,x4)
ginv(m1)
m2=t(m1)
ginv(m2)
Code:
> ginv(m1)
[,1] [,2] [,3] [,4]
[1,] 0.625 -0.45833333 0.04166667 0.3333333
[2,] -0.875 0.37500000 -0.12500000 1.0000000
[3,] 0.375 -0.20833333 0.29166667 -0.6666667
[4,] 0.750 -0.08333333 -0.08333333 -0.6666667
> m2=t(m1)
> ginv(m2)
[,1] [,2] [,3] [,4]
[1,] 0.62500000 -0.875 0.3750000 0.75000000
[2,] -0.45833333 0.375 -0.2083333 -0.08333333
[3,] 0.04166667 -0.125 0.2916667 -0.08333333
[4,] 0.33333333 1.000 -0.6666667 -0.66666667\(AGA=A\)
\(A^tG^tA^t=A^t\)
So Shall we restrict ourselves by saying that if G is the G inverse of A then\( G^t\) is one of the g inverse of \(A^t \)? or more generally we can say that
\((A^-)^t=(A^t)^-\)
just like ordinary inverses?Also any link for studying this will be helpful.
