Proof for certain matrix results?

Question

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:

1. 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 $B + \lambda AA^T \geq 0$
2. 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.

Answer 1

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.