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A random symmetric $2 \times 2$ matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22}\end{pmatrix}$ is a member of the gaussian orthogonal ensemble (GOE), if it satisfies three conditions:

$(1)$ for any $2 \times 2$ orthogonal transformation $OO^T = I = O^TO$, $A^{\prime} = OAO^T$ is also a member of the GOE.

$(2)$ the matrix elements $a_{11}, a_{12}$ and $a_{22}$ are statistically independent.

$(3)$ the probability density $P(A)dA$, where $dA=da_{11}da_{12}da_{22}$ is given by $$P(A) \propto e^{-aTrA^2 + bTrA + c}$$ where $a>0, b$ and $c$ are real numbers.

Question: Compare $P(A)$ above with the normal distribution $$P(x)= \dfrac{1}{\sqrt{2\pi \sigma^2}}{e^{-\frac{(x-\mu)^2}{2\sigma^2}}},$$ and explain the significance of $a,b$ and $c$.

I don't even know what is the meaning of $P(A)dA$, less to answer the question. Can anyone give some hint?

Idonknow
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$P(A) dA$ means $$ P( a_{11} < x, a_{12} < y , a_{22} < z ) = \underbrace{Const}_{\text{normilzation}} \int_{- \infty}^x \int_{- \infty }^y \int_{-\infty}^z e^{-a\text{Tr}(A^2)+b \text{Tr}(A) +c} da_{11}da_{12}da_{22} $$ I have a feeling they want you to do the following...first note that $$\text{Tr}(A^2) = a_{11}^2 +2a_{12}^2 + a_{22}^2 $$ $$\text{Tr}(A) = a_{11} + a_{22}$$ Using this, we see $$\exp(-a\text{Tr}(A^2)+b \text{Tr}(A)) = \exp (\underbrace{-aa_{11}^2 + b a_{11} }- \underbrace{a a_{22}^2 + ba_{22} } - \underbrace{2a a_{12}^2 } )$$ Now let's complete the square. We see if $\mu = b /2a$, we may write $$ \exp(-a\text{Tr}(A^2)+b \text{Tr}(A)) =\exp( -a(a_{11} -\mu)^2 - a(a_{22} - \mu)^2 -2a a_{12}^2 +2a\mu^2) $$ Now can you compare the first expression to a normal distribution using this modified form?

Jeb
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