Proof for certain matrix results? There are certain results of matrices that Stephen Boyd uses often in his book on Convex optimization. Can someone provide me proof for the results I have enumerated below:


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*If $B \in S^n$ and $A \in R^{p*n}$, then $x^TBx \geq 0$ for all $x \in N(A)$ if and only if there exist a $\lambda$ such that $B + \lambda AA^T \geq 0$

*For any $p$ that satisfies $Ax = b$, any solution to the equation $Ax = b$ can be written as $x = p + v$ where $v \in N(A)$


Besides the proof an analogy to $R$ or $R^2$ would be really helpful.
Thank you.

Q1: Sufficiency is seen by looking at $x^T(B+\lambda A^TA)x$. Necessity can be proved as follows:
Assume that, for every $\lambda$, there exists a nonzero $y$ such that $y^T(B+\lambda A^T A)y < 0$. $y \in N(A)$ would then imply $y^T B y <0$ (solution modified from "Matrix Mathematics", Fact 8.15.24).
Q2: $A(p+v) = Ap + Av = b + 0 = b$, since $v$ is in the nullspace.
