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seen Oct 10 at 14:29

quadratic linear form XFX with matrix

F:  1   1
    1   1

in canonical base can be expressed as X' F' X'

X'  1   0   X'   
    0   0

where X'=A-1X, F'=AFA and A is transition matrix. Canonical base vectors are [1,0] and [-1,1]. Thus this is correct that (a+b)^2 = a^2 + b^2 + 2ab


Oct
7
comment Prove an implication about quadratic form definiteness
@Daniel thank you very much
Oct
7
comment Prove an implication about quadratic form definiteness
@Daniel yes, that's it, you can post this as an answer, but I think we should first show that result is true for n=2, because for n=1 it is trivial. Then we show that it is true for any n, based on fact that n-1 is positive definite. The result for n comes always from the fact that $\det A^TA = ( \det A )^2>0$
Oct
7
comment Prove an implication about quadratic form definiteness
We want to prove => in statement: quadratic form $q$ over an $n$-dimensional real linear space $V$ is positive definite $\iff$ all main minors $\Delta_1, \Delta_2, ..., \Delta_n > 0$
Oct
7
comment Prove an implication about quadratic form definiteness
@Daniel "Thus, by induction on the dimension its main minors are positive." - but the fact that if q is positive then minors are positive is what we have to PROVE now
Oct
6
revised Prove an implication about quadratic form definiteness
refactor explanation added
Oct
5
revised Prove an implication about quadratic form definiteness
refactor
Oct
5
revised Prove an implication about quadratic form definiteness
refactor grammar
Oct
5
revised Prove an implication about quadratic form definiteness
refactor explanation
Oct
5
asked Prove an implication about quadratic form definiteness
Jul
2
awarded  Curious
Jun
29
comment Betfair Odds Percentage Movements & Hedging
@user2096512 what do you mean by hedged profit?
Jun
8
answered Why is one of the conditions of a vector space that if I add two vectors, the sum must be within the space?
May
15
accepted dimension of the sum of two planes in $\mathbb{R}^3$
Apr
29
comment dimension of the sum of two planes in $\mathbb{R}^3$
I have upvoted this answer, though it is hardly complete, because it gave me a direction
Apr
29
revised dimension of the sum of two planes in $\mathbb{R}^3$
add explanation
Apr
29
revised dimension of the sum of two planes in $\mathbb{R}^3$
refactor
Apr
29
revised dimension of the sum of two planes in $\mathbb{R}^3$
refactor
Apr
29
revised dimension of the sum of two planes in $\mathbb{R}^3$
refactor
Apr
29
revised dimension of the sum of two planes in $\mathbb{R}^3$
refactor
Apr
29
asked dimension of the sum of two planes in $\mathbb{R}^3$