Can we think that any matrix representation has an underlying co-ordinate system? Now consider a positive definite sample covariance matrix. If so what is the meaning of the co-ordinate system of the covariance matrix?
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Not sure what you mean exactly, but perhaps this will help. – AnonSubmitter85 Apr 17 '15 at 23:11
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Not sure if this exactly answers your question, but any positive definite symmetric matrix (e.g. covariance matrix) can be written as $A^TA$ for some square matrix $A$. Then, when you consider the covariance between two vectors $x,y$ you get $x^T(A^TA)y = (Ax)^T(Ay)$, so the structure of the covariance can be considered to be the covariance between linear combinations of standard independent variables. And $A$ gives the linear combinations (i.e., the coordinate system).
user2566092
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Thank you for your reply. Do you mean the co-ordinate system means the values of the variable? – Creator Apr 17 '15 at 23:09
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No, I mean the columns of $A$ define the coordinate system, i.e. change of basis, that allow you to view your covariance as covariance between independent variables. This is important for example when you want to sample from an arbitrary Gaussian with arbitrary covariance matrix, given that you can sample from a Gaussian with identity covariance matrix (which is easy by sampling each coordinate from a regular one-dimensional Gaussian). – user2566092 Apr 17 '15 at 23:12
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thank you. May be my question is not correct. Any column of a matrix may be considered as a change of basis, I was thinking what the basis itself means not change of basis. For example for our usual 3 dimension we can think of the basis as distance in a direction. Sorry if my question makes no sense. – Creator Apr 17 '15 at 23:19