For matrices $2 \times 2$, conditions for $A$ such that $tr(XAY^{T})$ is an inner product

Question

Given a generic $2 \times 2$ matrix $A$:

$$ \left[ \begin{array}{cc} a&b\\ c&d \end{array} \right] $$

the exercise is to determine conditions for $a$, $d$ and $b-c$ such that $tr(XAY^{T})$ is an inner product.

Verifying linearity and symmetry we get $b=c$.

My problem is verifying the positive-definite propertie. I arrived at the expression, where $x_1, x_2, x_3, x_4$ are the elements of the matrix $X$:

$tr(XAX^{T}) = a(x_1^2+x_3^2) + d(x_2^2+x_4^2) + (b+c)(x_1x_2 + x_3x_4)$

but was unable to analyse the conditions $a$ and $d$ should satisfie.

Answer 1

For the positive-definite part, suppose that $A$ is not positive definite. Then there exists nonzero $z\in\mathbb R^2$ such that $t=z^TAz\leq0$. So $$ \operatorname{Tr}(zz^TAzz^T)=t\,\operatorname{Tr}(zz^T)\leq0, $$ since $\operatorname{Tr}(zz^T)\geq0$.

Answer 2

Completing the square should help here. For instance, \begin{align*} ax_1^2 + d x_2^2 + 2bx_1x_2 &= a\left(x_1 + \frac{b}{a}x_2\right)^2 + \left(d - \frac{b^2}{a}\right)x_3^2 \end{align*} You want the coefficients of the squares on the right to be positive.

Answer 3

$tr(XAX^{T}) = a(x_1^2+x_3^2) + d(x_2^2+x_4^2) + (b+c)(x_1x_2 + x_3x_4)$

$tr(XAX^{T}) = a(x_1^2+x_3^2) + d(x_2^2+x_4^2) + (2b)(x_1x_2 + x_3x_4)=PAP^T+QAQ^T$

where $P=(x_1,x_2)$ and $Q=(x_3,x_4)$

For $tr(XAX^{T})\ge 0$ you need both the terms on right side individually $\ge 0$ for which you must have $A$ to be positive definite. Using Sylvester criterion of positive definite:

$a>0$ and $ad-b^2>0\implies d>b^2/a>0$