On Jun 6, 9:58 am, achille <achille_...@yahoo.com.hk> wrote: > On Jun 6, 7:49 pm, Ron <ron.sper...@gmail.com> wrote: > > > > > > > On Jun 5, 8:39 am, achille <achille_...@yahoo.com.hk> wrote: > > > > On Jun 5, 4:10 am, Kaba <n...@here.com> wrote: > > > > > achille wrote: > > > > > On Jun 4, 9:48 pm, Kaba <n...@here.com> wrote: > > > > > > Hi, > > > > > > > This is part of the first question for chapter 1 of "Applied numerical > > > > > > linear algebra" book, but I just can't come up with a solution: > > > > > > > If A and B are orthogonal matrices, and det(A) = -det(B), show that > > > > > > A + B is singular. > > > > > > > Any hints? > > > > > > > --http://kaba.hilvi.org > > > > > > Hint: A^t (A+B) B^t = B^t + A^t and take det(.) on both sides. > > > > > Well, that's straightforward, thanks:) > > > > > I am not sure if this is a best kind of exercise, at least when > > > > separated from any context. It seems this is just a trick, with no > > > > deeper lesson to learn. But maybe it is used somewhere in the next > > > > pages. > > > > > --http://kaba.hilvi.org > > > > How about this proof: > > > > Let S be the unit sphere and for 0 <= t <= 1, S(t) be > > > S's image under the linear map x -> ((1-t) A + t B) x. > > > The volume of S(0) is det(A), the volume of S(1) is det(B). > > > Since det(A) = -det(B), there is a 0 < t < 1 such that > > > volume of S(t) = 0. ie. det( (1-t)A + tB ) = 0. > > > For such t, there is a non-zero vector y such that > > > ((1-t)A + tB) y = 0 > > > => (1-t)A y = -tB y, > > > => (1-t) |y| = (1-t)|Ay| = t |B y| = t |y| > > > => 1-t = t > > > => t = 1/2 > > > => det((A+B)/2) = 0. > > > I'm not sure what you mean by "volume" here as volume as generally > > defined is a positive number and you are arguing that either S(0) or > > S(1) has negative volume. > > It simply refers to the fact the two embedding S(0) and S(1) of the > sphere (to be precise, the n-dim ball which is an orientable n- > manifold) > of unit volume into R^n has different orientations. The orientation of > one of S(0) and S(1) will be opposite to the standard orientation of > R^n.
This still doesn't say how you are defining volume here. I'm not trying to knock down your argument since I see what you are trying to do here, but what is your definition of volume? Absolutely A and B give different orientations and I think I see what your argument is, but I think if you are going to use "volume" in a sense that isn't some kind of measure, then it needs to be defined.