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# Gaussian Copula and the addition onofan Identity matrix

When I was looking at the Gaussian Copula Example @ http://en.wikipedia.org/wiki/Copula_(probability_theory)

I realized the Gaussian Copula is stated as follow

$$C^{Gauss}_\Sigma \begin{equation} C^{Gauss}_\Sigma (u) = \frac{1}{\sqrt\det{\Sigma}} \exp{\Bigg ( -\frac{1}{2} \begin{pmatrix} \Phi^{-1}(u_1) & \ \dots & \ \Phi^{-1}(u_d)\end{pmatrix}^T. (\Sigma^{-1} - I).\begin{pmatrix} \Phi^{-1}(u_1) & \ \dots & \ \Phi^{-1}(u_d)\end{pmatrix} \Bigg) }$$\end{equation} where $\Sigma$ is the correlation matrix, $\Phi^{-1}$ is the inverse cumulative distribution function of a standard normal and $I$ is the identity matrix.

The question is, why is there an identity matrix in the exponential form?

Thank you

1

# Gaussian Copula and the addition on Identity matrix

When I was looking at the Gaussian Copula Example @ http://en.wikipedia.org/wiki/Copula_(probability_theory)

I realized the Gaussian Copula is stated as follow

$$C^{Gauss}_\Sigma (u) = \frac{1}{\sqrt\det{\Sigma}} \exp{\Bigg ( -\frac{1}{2} \begin{pmatrix} \Phi^{-1}(u_1) & \ \dots & \ \Phi^{-1}(u_d)\end{pmatrix}^T. (\Sigma^{-1} - I).\begin{pmatrix} \Phi^{-1}(u_1) & \ \dots & \ \Phi^{-1}(u_d)\end{pmatrix} \Bigg) }$$

where $\Sigma$ is the correlation matrix, $\Phi^{-1}$ is the inverse cumulative distribution function of a standard normal and $I$ is the identity matrix.

The question is, why is there an identity matrix in the exponential form?

Thank you