Matrix A + B -- Typesetting tip
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From: JEMebius on 5 Jun 2010 17:28 Kaba wrote: > Ron wrote: >> On Jun 5, 7:23 am, Kaba <n...(a)here.com> wrote: >>> Stephen Montgomery-Smith wrote: >>>> Kaba 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? >>>> Here is my attempt. If A is orthogonal, then the determinant is either >>>> 1 or minus 1. WLOG det(A) = 1. Now >>>> A+B = A(I + A^{-1} B) >>>> so WLOG A=I, and B is an orthogonal matrix whose determinant is -1. >>> Could you be more specific why WLOG here? >>> >> I suspect the idea is this. A is invertible so A+B is singular if and >> only if I+A^{-1}B is singular. So we simply consider the case where >> A=I and B is an orthogonal matrix of determinant -1. The argument is >> essentially the same as the earlier one using the fact that an >> orthogonal matrix of determinant -1 has -1 as an eigenvalue. > > Ok, I see, it works:) > Typesetting tip: A^-1 instaed of A^(-1) The Thunderbird EMail/Newsgroup program interprets "^" (caret) as exponentiation. I guess all such programs do so. The string following "^" is converted to superscript as far as it looks like a number, positive or negative, integer or non-integer. My guess: any character other than 0 1 2 3 4 5 6 7 8 9 + - . stops the superscript mode. Check: A^0, A^+1, A^1, A^+1.2345, A^1.2345, A^-6, A^-6.7890, A^.5, A^-.5 Johan E. Mebius |