Symmetric Matrixes Thm

blue_m

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.

gtse

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*

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.

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blue_m	Symmetric Matrixes Thm Tweet I need to prove the following. 1. A is a symmetric matrix, and x(transpose)Ax=0 for all x (belongs to R^n) if and only if A=0. 2. x(transpose)Ax=0 for all x (belongs to R^n), if and only if A is skew symmetric.

gtse	I can suggest that for the second one that you transpose both sides and use the A(transposed) = -A property so that x(transpose)Ax = 0 = -(x(tranpose)Ax) so those two can equal 0 if and only if A is 0

trambolin	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.