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 December 8th, 2015, 09:21 PM #1 Newbie   Joined: Oct 2014 From: China Posts: 12 Thanks: 2 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 09:25 PM. February 17th, 2016, 07:49 AM #2 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 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. February 19th, 2016, 04:50 AM #3 Senior Member   Joined: Feb 2012 Posts: 144 Thanks: 16 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). Tags –, covariance, diagonalization, hermitian, inverse, matrix, operator, problem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ulrichthegreat Linear Algebra 1 October 6th, 2014 12:51 AM david940 Linear Algebra 0 June 29th, 2014 04:32 AM phd_student Linear Algebra 0 January 1st, 2013 12:53 AM diegomoreno Real Analysis 0 November 28th, 2012 01:42 PM excellents Linear Algebra 0 October 17th, 2009 08:12 AM

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