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Consider a function $f(\mathbf{x})=\mathbf{M}_\mathbf{x}$ that outputs a nonsymmetric matrix $\mathbf{M}_\mathbf{x} \in \mathbb{R}^{N \times N}$ given an input vector $\mathbf{x} \in \mathbb{R}^N$.

Is the following condition possible without requiring a positive definite $\mathbf{M}_\mathbf{x}$? I think so... $$ \mathbf{x}^T f(\mathbf{x}) \, \mathbf{x} = \mathbf{x}^T \, \mathbf{M}_\mathbf{x} \, \mathbf{x} > 0 \hspace{0.4cm} \forall \mathbf{x}, \mathbf{x} \neq \mathbf{0} $$ And how can I prove this statement without showing that $\mathbf{M}_\mathbf{x}$ is always positive definite?

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Yes, it is possible; pick a nonzero element of $\mathbf{x}$, which must exist since $\mathbf{x}\neq 0$; let's say this nonzero element is $x_j$. Then define $$\big(\mathbf{M}_{\mathbf{x}}\bigr)_{nm}=-\delta_{nm}(1-\delta_{nj})+\delta_{nm}\delta_{nj}\left(x_{j}^{-2}+x_{j}^{-2}\sum_{k\neq j}x_k^2\right).$$ Now you have the desired result, $$\sum_{n,m}x_nx_m\big(\mathbf{M}_{\mathbf{x}}\bigr)_{nm}=1>0,$$ while $\mathbf{M}_{\mathbf{x}}$ is itself not positive definite.

This matrix is symmetric, if you want it to be nonsymmetric, simply add to $\mathbf{M}_{\mathbf{x}}$ an arbitrary skew-symmetric matrix $\mathbf{A}$ and use that $\mathbf{x}^T (\mathbf{M}_{\mathbf{x}}+\mathbf{A})\mathbf{x} =\mathbf{x}^T \mathbf{M}_{\mathbf{x}}\mathbf{x}$.

I don't understand your second question "And how can I prove this statement without showing that $\mathbf{M}_{\mathbf{x}}$ is always positive definite?" To prove this you need to examine the function $f(\mathbf{x})$, without further knowledge of this function no progress can be made.

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