Symmetric Matrixes Thm


by blue_m
Tags: hermitian, matrices, skew, symmetric
blue_m
blue_m is offline
#1
Apr25-10, 02:51 AM
P: 1
I need to prove the following.
1. A is a symmetric matrix, and x(transpose)*A*x=0 for all x (belongs to R^n) if and only if A=0.
2. x(transpose)*A*x=0 for all x (belongs to R^n), if and only if A is skew symmetric.
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
gtse
gtse is offline
#2
Apr25-10, 10:52 PM
P: 5
I can suggest that for the second one that you transpose both sides
and use the A(transposed) = -A property so that
x(transpose)*A*x = 0 = -(x(tranpose)*A*x)
so those two can equal 0 if and only if A is 0
trambolin
trambolin is offline
#3
Apr27-10, 03:55 AM
P: 341
For the first one, sufficiency is obvious. For necessity, either, write the explicit quadratic polynomial and plug in all values of x. Only possibility is the zero polynomial. Alternatively, Assume your [itex]A=V\Lambda V^{-1}[/itex] is diagonalizable with some Jordan blocks (possibly with size [itex]k \times k[/itex]). And use the structure of the Jordan blocks.


Register to reply

Related Discussions
Comparison of matrixes General Math 5
dimension of symmetric and skew symmetric bilinear forms Linear & Abstract Algebra 2
micro-matrixes resources Biology 0
Matrixes help Linear & Abstract Algebra 4