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I have three questions to ask, hope to get some help from you (1) Is there any effective way to determine the negative definiteness of a N-order matrix $\textbf{M}$? I think it's impossible to solve it by using the order principal minor determinant or eigenvalue method of $-\textbf{M}$, so how can I prove it? thx! $\textbf{M}$ is a symmetric matrix.

(2) If $\frac{\partial \mathcal{F}}{\partial B} = {A^{\dagger}} \left( \Omega+ A B A^{\dagger} \right)^{-1} A\\$ ($A, B, \Omega$ are all N-order matrices), Then how to calculate the second derivative of B $\frac{\partial^2 \mathcal{F}}{\partial B^2}$? Should it be $\frac{\partial^2 \mathcal{F}}{\partial B^2} = {A^{\dagger}}{A^{\dagger}} \left( \Omega+ A B A^{\dagger} \right)^{-2} AA\\$ or according to $\frac{d}{dx}A^{-1}=-A^{-1}\frac{dA}{dx}A^{-1}$, we can get $\frac{\partial^2 \mathcal{F}}{\partial B^2} = -{A^{\dagger}}{A^{\dagger}}\left( \Omega+ A B A^{\dagger} \right)^{-1} \left( \Omega+ A B A^{\dagger} \right)^{-1} AA\\$? Or there is any other way to solve it? thx!

(3) Game theory question: How to present the best response mapping of a two-player pure strategic game? I mean, if the payoff function $x_1^* = \arg \max_{x_1} u_1(x_1,x_2)=\log(\frac{1}{1+x_1+x_2})-\lambda_1 x_1$ and from symmetrics we can get the $\arg \max_{x_2} u_2(x_2,x_1)=\log(\frac{1}{1+x_1+x_2})-\lambda_2 x_2$ are all concave functions, but we know that depending on the value range of $x$ and $\alpha$, we may have 3 cases for the maximum of $u_1, u_2$, then how can we write down the best response mapping? Are best response mapping and best response function the same definitions?

Thank you very much for your help!

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It is better to send $3$ different posts.

  1. You want $M<0$, that is $-M>0$. You can calculate the $N$ $\det$ of the principal minors of $-M$; numerically the complexity is in $O(N^4)$. There is a better method: apply to $-M$ the Choleski algorithm; if it works, then $-M>0$, otherwise it is not. Moreover the complexity of this algorithm is in $O(N^3)$.

  2. To calculate $\frac{\partial^ 2 \mathcal{F}}{\partial B^2}$ is not difficult if your $\frac{\partial \mathcal{F}}{\partial B}$ is correct. Indeed, I do not understand how you found this first derivative; in any case, $\mathcal{F}$ is certainly not a $\log$ function.

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    $\begingroup$ Thank you! I wil modify this $\endgroup$ – Jie Tian May 12 '15 at 20:23
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Answer to (2): In general derivative of a matrix valued function by a matrix will give a 4-dimensional array. We have $\partial(a^{-1})_{ij}/\partial a_{kl}=-(a^{-1})_{ik}(a^{-1})_{lj}$. We this, chain and product rule it should be poosible to compute the derivative. However it requires some lengthy computations.

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  • $\begingroup$ Thanks for your answer! But i think this is just the definition of derivative of matrix. e.g. $F(X) = (f_{ij}(X))_{s \times t}, X = (x_{ij})_{m \times n}$, then $\frac{dF}{dX} \in C^{ms \times nt}$, which is a super matrix right? but I wondered if there is any theorem or formula so I can calculate more easily, since it's matrix valued function, we can treat the matrix element as a whole and do not pay much attention to the entries inside? thx! and is there anybody who has any idea about the other two questions? thx! $\endgroup$ – Jie Tian Apr 8 '15 at 18:55

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