Prove Symmetric Matrixes Thm: A=0 or Skew Symmetric

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This discussion focuses on proving two theorems related to symmetric matrices. The first theorem states that a symmetric matrix A satisfies the condition x(transpose)*A*x=0 for all x in R^n if and only if A=0. The second theorem asserts that if x(transpose)*A*x=0 for all x in R^n, then A must be skew symmetric. The proof for the second theorem involves using the property A(transposed) = -A and analyzing the implications of the quadratic polynomial formed by the expression.

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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.
 
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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
 
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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