Useful Matrix properties

Theorem

Let \(A\) be \(m\times n\) matrix, \(B\) and \(C\) \(n\times n\) matrices, with \(B\) symmetric, \(x\) \(n\times 1\) vector

1. \(Ax=0 \; \forall x\) if and only if \(A=0\),
2. \(x^TAx \; \forall x\) if and only if \(B=0\),
3. \(x^TCx \; \forall x\) if and only if \(C^T=-C\) (skew symmetric).