$2×2 $ real symmetric matrix full classification

Question

Let $$A=\begin{pmatrix}a&b\\b&c\end{pmatrix}$$ all real numbers.

I know that

• If $\operatorname{det}(A) > 0$ then $A$ is positive definite if $a > 0$ and negative definite if $a < 0$.
• If $\operatorname{det}(A) < 0$ then $A$ is indefinite.
• If $ \operatorname{det}(A) = 0 $ and $ a > 0 $, then $ A $ is positive semidefinite.
• If $ \operatorname{det}(A) = 0 $ and $ a < 0 $, then $ A $ is negative semidefinite.
• If $ \operatorname{det}(A) = 0 $, $ a = 0 $, and $c>0$ then $ A$ is positive semidefinite.
• If $ \operatorname{det}(A) = 0 $, $ a = 0 $, and $c<0$ then $ A$ is negative semidefinite.
• If $ \operatorname{det}(A) = 0 $, $ a = 0 $, and $c=0$ then $ A$ is the zero matrix and then is positive semidefinite and negative semidefinite.

Is this classification correct and complete? Why this classification is not in every textbook about quadratic forms?

Answer 1

$A=\left[\begin{array}{cc}x+y&z\\z&x-y\end{array}\right]$ is also an interesting parameterization, with the cone of revolution $x>\sqrt{y^2+z^2}$ visualizing the positive definite matrices, and so on. Another advandage is the invariance of the classification by the rotations around the $Ox$ axis, and a geometrical view of the space of symmetric matrices of order 2. .