Cauchy-Schwarz Inequality for a Positive Semidefinite Quadratic Form on $\mathbb{R}^n$

Let $n$ be a natural number with $1\le n$, let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, and let $B$ be a symmetric positive semidefinite real $n\times n$ matrix. On Euclidean space $\mathbb{R}^n$, a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, write $x\cdot y$ for the dot product, $Bx$ for the matrix-vector product, and $0_{\mathbb{R}^n}$ for the origin. For $t\in\mathbb{R}$ write $t^{2}$ for $t\,t$. Let $x,y\in\mathbb{R}^n$.

Then the following hold.

1. (Symmetry of the form) $y\cdot(Bx)=x\cdot(By)$.

2. (Cauchy-Schwarz)

$$\bigl(x\cdot(By)\bigr)^{2}\le\bigl(x\cdot(Bx)\bigr)\,\bigl(y\cdot(By)\bigr).$$

3. (Null vectors) If $x\cdot(Bx)=0$, then $Bx=0_{\mathbb{R}^n}$.