Meaning of co-ordinate system of Covariance matrix

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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 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).