for part B: note that:
see if you can derive a similar expression for . this should give you 2 positive terms, and 2 negative terms, for a total of 4.
3)
A)
there is a lenear transformation T:V->V in inner product
it is given that . prove that T is positive definite and find every posible
eigenvalue of T
?
B)
we define
write the formula as a sum of squares,find the rank and signature of q
?
regarding A:
the book definition is:
a matrices which is hermitic A^{*}=A and for every vector and k of our choosing
i dont know how to use it here
regarding B :
in the books definition a signature is \sigma=\pi-\mu
is the number of positive members=2
is the number of negative members =4
i got
??
in order to find the rank i need to find representative matrices in
and the rest stays the same
i got here 5 member in power 2
but my matrices is 4x4 i cant have 5 members in the diagonal of 4x4 matrices
??
think of it like "a change of variables". the set of variables {u1 = x1+x4, u2 = -x1, u3 = -x4, u4 = x2-3x3, u5 = -x2, u6 = -9x3} is not linearly independent,
whereas {u1 = x1+x4, u2 = x1-x4, u3 = x2+x3, u4 = x2-x3} is linearly independent.
i used a method in my book called lagrange method.
it tells to complete the members into the (a+b)^2 formula
i see that my variables are dependant but i coulnt think of another way of solving it
is there a general method of solving this type of questions without getting a dependant set of variables in the end
?
is -x1^2 - x2^2 of either form (a+b)^2 or (a-b)^2? no, it is not.
however: (a+b)^2 - (a-b)^2 = 4ab, that is:
ab = ((a+b)/2)^2 - ((a-b)/2)^2 and these terms are of the required form (using a' = a/2, b' = b/2).
if you have a term like:
kab, then you use a' = a√k/2, b' = b√k/2.
your way gves you too many variables to be able to come up with a 4x4 matrix.
the goal is to replace each variable x1,x2,x3,x4 with some other variables u1,u2,u3,u4,
so that q(u1,u2,u3,u4) = ±(c1u1)^2 ± (c2u2)^2 ± (c3u3)^3 ± (c4u4)^2, so that the matrix for q
in the variables (u1,u2,u3,u4) is diag(±c1,±c2,±c3,±c4).