# Jacobian and covariance matrix

Anyone know the Bishop's book in 2.53 they use Jacobian to convert covariance matrix x to y.

$$J_{ij}=\dfrac{\partial x_i}{\partial y_i}=U_{ji} \qquad{(2.53)}$$ $$\int_{\bf x} f({\bf x})d{\bf x} = \int_{\bf y} f({\bf y})|{\bf J}|d{\bf y}$$ $${\vert}{\bf J}{\vert}^2 = {\vert}{\bf U}^T{\vert}^2 = {\vert}{\bf U}^T{\vert}\;{\vert}{\bf U}{\vert} = {\vert}{\bf U}^T{\bf U}{\vert} = {\vert}{\bf I}{\vert} = 1 \qquad{(2.54)}$$ $$\left|\Sigma\right|^{\frac{1}{2}}=\prod_{j=1}^{D}\lambda_j^{\frac{1}{2}} \qquad{(2.55)}$$

$$p({\bf y}) = p(x)|{\bf J}| = \prod_{j=1}^{D}\dfrac{1}{(2\pi\lambda_j)^{1/2}}\exp\left\{-\dfrac{y_j^2}{2\lambda_j}\right\} \qquad{(2.56)}$$

1) why they need to change it with Jacobian ?(what is the reason) 2) what is the p(y) represent in 2.56 3) why they use jacobian for in machine learning covariance matrix?

2) what is the p(y) represent: If we perform the following linear transformation: $$y=U^{T}(x -u)$$
Where U is the eigenvectors of the covariance matrix. The new variable y will be zero mean and uncorrelated i.e: $$p(y) = N(0,diag(λ_1,..,λ_N))$$
$$p(y_1,..,y_N)=p(y_1)p(y_2)..p(y_{N-1})p(y_N)$$