Q1: Which matrix $A \in \mathbb{C}^{2 \times 2}$ is clearly not positive definite?

A1: $ A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix} $

A2: $ A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix} $

A3: $ A = \begin{pmatrix} 5 & 1 \ 1 & 1 \end{pmatrix} $

A4: $ A = \begin{pmatrix} 2 & 3 \ 3 & 5 \end{pmatrix} $

Q2: What is not equivalent for $A = \begin{pmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{pmatrix} \in \mathbb{C}^{2 \times 2}$ being a positive-definite matrix?

A1: $a_{11} > 0$ and $ \det(A) > 0$.

A2: $A$ is selfadjoint and $\langle x, A x \rangle_{standard} > 0 $ for all $x \in \mathbb{C}^2 \setminus { 0 } $.

A3: $A$ is selfadjoint and all eigenvalues are positive.

A4: $\langle x, A x \rangle_{\text{standard}} > 0 $ for all $x \in \mathbb{C}^2 \setminus { 0 } $.

Q3: Apply a Gaussian step to the matrix $A = \begin{pmatrix} 2 & 1 \ 1 & 1 \end{pmatrix}$ that is allowed according to the video such that one can check if $A$ is positive definite. Which matrix do we get?

A1: $A = \begin{pmatrix} 2 & 1 \ 0 & \frac{1}{2} \end{pmatrix}$

A2: $A = \begin{pmatrix} - 1 & -\frac{1}{2} \ 1 & 1 \end{pmatrix}$

A3: $A = \begin{pmatrix} 1 & 1 \ 2 & 1 \end{pmatrix}$

A4: $A = \begin{pmatrix} 2 & 1 \ -2 & -2 \end{pmatrix}$