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December 8th, 2015, 10:21 PM   #1
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Inverse problem of covariance matrix – diagonalization of Hermitian operator

(I had posted the question elsewhere but got no reply)

I have understood the two things respectively:
1. Use a set of observations to construct a covariance matrix, and then compute the eigenvectors of the matrix.
2. The diagonalization the Hermitian operator $A=PGP^T$. The columns of $P$ are eigenvectors. The diagonals of $G$ are eigenvalues.

However, how the two processes are related?
Consider a toy problem, or an inverse problem of covariance matrix:
A covariance matrix is given:
$$
\left(\begin{array}{ccc}
4 & -2 & 0 & 0 \\
-2& 4& -2 & 0\\
0 & -2 & 4 & -2\\
0 & 0 & -2 & 4\\
\end{array}\right)
$$
Can we figure out the possible observations that lead to this covariance matrix? One possible solution is:

1st observation: $(1,1,1,1)$

2nd observation: $(\frac{-1-\sqrt{3}}{2},\frac{-1+\sqrt{3}}{2},\frac{-1+\sqrt{3}}{2},\frac{-1-\sqrt{3}}{2})$

3rdobservation $(\frac{-1+\sqrt{2}}{3},\frac{1+\sqrt{2}}{3},\frac{1-\sqrt{2}}{3},\frac{-1-\sqrt{2}}{3})$

4th observation: $(-\frac{\sqrt{3}}{2\sqrt{2}}+\frac{\sqrt{10}}{4},-\frac{\sqrt{3}}{2\sqrt{2}}-\frac{\sqrt{10}}{4},\frac{\sqrt{3}}{2\sqrt{2}}+\fr ac{\sqrt{10}}{4},\frac{\sqrt{3}}{2\sqrt{2}}-\frac{\sqrt{10}}{4})$

The pattern resembles the Fourier series. That’s why I am wondering how the construction of covariance matrix and the matrix diagonlization are somehow connected.

On the other hand, one can write $A=PGP^T$ as follows:
$$
A=\left(\begin{array}{cc}
x_1 & x_2 & x_3 \\
y_1 & y_2 & x_3 \\
z_1 & z_2 & z_3 \\
\end{array}\right)
\left(\begin{array}{cc}
\lambda_1 & 0 & 0 \\
0 & \lambda_2 & 0 \\
0 & 0 & \lambda_3 \\
\end{array}\right)
\left(\begin{array}{cc}
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
x_3 & y_3 & z_3 \\
\end{array}\right)\\
=\left(\begin{array}{cc}
\lambda_1 x_1^2+\lambda_2 x_2^2 +\lambda_3 x_3^2& \lambda_1 x_1y_1+\lambda_2 x_2y_2 +\lambda_3 x_3y_3 & \lambda_1 x_1z_1+\lambda_2 x_2z_2 +\lambda_3 x_3z_3 \\
\lambda_1 x_1y_1+\lambda_2 x_2y_2 +\lambda_3 x_3y_3 & \lambda_1 y_1^2+\lambda_2 y_2^2 +\lambda_3 y_3^2 & \lambda_1 y_1z_1+\lambda_2 y_2z_2 +\lambda_3 y_3z_3 \\
\lambda_1 x_1z_1+\lambda_2 x_2z_2 +\lambda_3 x_3z_3 & \lambda_1 y_1z_1+\lambda_2 y_2z_2 +\lambda_3 y_3z_3 & \lambda_1 z_1^2+\lambda_2 z_2^2 +\lambda_3 z_3^2 \\
\end{array}\right)
$$

Can one interpret these values as observations in a context of covariance?

Last edited by whitegreen; December 8th, 2015 at 10:25 PM.
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February 17th, 2016, 08:49 AM   #2
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I am not sure what you are asking. The problem you post says "Use a set of observations to construct a covariance matrix, and then compute the eigenvectors of the matrix" but you start with the covariance matrix and try to go back to a "set of observations". I don't believe that is possible. There may be many different "sets of observations" that give the same covariance matrix.
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February 19th, 2016, 05:50 AM   #3
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It is easy to generate a sample having approximately as variance matrix a given matrix (provided that this matrix is symmetric and definite positive). It A is this matrix, just write the Choleski decomposition A=HH' , then generate a standard multivariate gaussian sample of size n, so that you get an nxd matrix X (assuming A is dxd) and put Y=XH. Then VarY is approximately A (converges to A when n tends to infinity).
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