Is there a unique matrix $K \in \mathbb R^{2 \times 2}$ such that $KA + B$ is positive definite ($A$, $B$ given below)

Question

Is there a unique matrix $K \in \mathbb R^{2 \times 2}$ (up to multiplication by a constant factor) such that $$K \begin{pmatrix} 0 &-1 \\ -1 & 0 \end{pmatrix} + \begin{pmatrix} 0 &0 \\ 0 & -1 \end{pmatrix} $$ is a positive definite matrix?

Answer 1

If $K = M\begin{bmatrix} 0 & -1 \\ -1 & 0\end{bmatrix}$ then $$K \begin{bmatrix} 0 & -1 \\ -1 & 0\end{bmatrix} + \begin{bmatrix}0&0\\0&-1\end{bmatrix} = M + \begin{bmatrix}0&0\\0&-1\end{bmatrix}.$$

There are many choices of $M$ that will work, like $\begin{bmatrix} c_1 & 0 \\ 0 & 1 + c_2\end{bmatrix}$ for any positive $c_1, c_2$. This corresponds to $K = \begin{bmatrix} c_1 & 0 \\ 0 & 1 + c_2\end{bmatrix} \begin{bmatrix} 0 & -1 \\ -1 & 0\end{bmatrix}$ which are not scalar multiples of each other.