- #36
snoble
- 127
- 0
But it does it in a predictable way. So if it switches the sign then the small matrix you added to give a negative determinant will actually give a positive determinant after you undo the row reduction. But the small matrix that you added to give a positive determinant fixes that. To formallize a little take P=diag{1,1,1,...,1} and N=diag{1,1,1,...,-1}. Let S be your singular matrix and let A be the invertable matrix such that AS is upper triangular with only non-negatives on the diagonal. Then for t>0,
det(AS+tN)<0<det(AS+tP).
And so [tex]det(A^{-1})\cdot det(AS+tN)=det(S + tA^{-1}N)\ne 0, det(A^{-1})\cdot det(AS+tP)=det(S + tA^{-1}P)\ne 0 [/tex] and they have opposite signs. But t can be arbitraly small... small enough so that [tex]S + tA^{-1}N[/tex] is inside an epsilon ball of S. Ok so not that formal.
Bah I probably should have just kept my nose out of your guys conversation.
Cheers,
Steven
det(AS+tN)<0<det(AS+tP).
And so [tex]det(A^{-1})\cdot det(AS+tN)=det(S + tA^{-1}N)\ne 0, det(A^{-1})\cdot det(AS+tP)=det(S + tA^{-1}P)\ne 0 [/tex] and they have opposite signs. But t can be arbitraly small... small enough so that [tex]S + tA^{-1}N[/tex] is inside an epsilon ball of S. Ok so not that formal.
Bah I probably should have just kept my nose out of your guys conversation.
Cheers,
Steven