$v=\left(\!\!\begin{array}{c}x\\y\\z\end{array}\!\!\right)$ , $w=\left(\!\!\begin{array}{c}w^1\\w^2\\w^3\end{array}\!\!\right)$

• $\left(\begin{array}{ccc}a&0&0\\ 0&b&0\\ 0&0&c\end{array}\right)$ , $\left(\begin{array}{ccc}a&0&0\\ 0&b&0\\ 0&0&b\end{array}\right)$ , $\left(\begin{array}{ccc}a&0&0\\ 0&a&0\\ 0&0&a\end{array}\right)$

$[x,y,z]\left[\begin{array}{ccc}a&0&0\\ 0&b&0\\ 0&0&c\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]=ax^2+by^2+cz^2\ge0$

if and only if

$a,b,c\ge0$

• $\left(\begin{array}{ccc}a&0&0\\ 0&b&1\\ 0&0&b\end{array}\right)$ , $\left(\begin{array}{ccc}a&0&0\\ 0&a&1\\ 0&0&a\end{array}\right)$

$[x,y,z]\left[\begin{array}{ccc}a&0&0\\ 0&b&1\\ 0&0&b\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]$

$=ax^2+by^2+yz+cz^2=ax^2+b(y+\frac{z}{2b})^2+(b-\frac{1}{4b^2})z^2\ge0$

if and only if

$a\ge0$ , $b\ge\frac{1}{\sqrt[3]4}$

• $\left(\begin{array}{ccc}a&1&0\\ 0&a&1\\ 0&0&a\end{array}\right)$
• $[x,y,z]\left[\begin{array}{ccc}a&1&0\\ 0&a&1\\ 0&0&a\end{array}\right]\left[\begin{array}{c}x\\y\\z\end{array}\right]$

$=ax^2+xy+ay^2+yz+az^2$

$=a(x+\frac{y}{2a})^2+(a-\frac{1}{2a^2})y^2+a(z+\frac{y}{2a})^2\ge0$

if and only if

$a\ge\frac{1}{\sqrt[3]{2}}$

• $\left(\begin{array}{ccc}a&0&0\\ 0&b&c\\ 0&-c&b\end{array}\right)$ , $\left(\begin{array}{ccc}a&0&0\\ 0&a&b\\ 0&-b&a\end{array}\right)$ , $\left(\begin{array}{ccc}a&0&0\\ 0&a&a\\ 0&-a&a\end{array}\right)$