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  1. A matrix can be interpreted as the representation of a linear mapping between two vector spaces under their chosen bases, the Gram matrix of a bilinear form on two vector spaces, and possibly other kinds of interpretation I don't know yet?
  2. I was wondering how to interpret a normal matrix (i.e. a square matrix $A$ s.t. $A^* A=AA^*$) in vector spaces?
  3. What kind of linear mappings is represented as a positive definite matrix under some possibly special basis?

Thanks and regards!

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    What's the question in 1? That seems like a statement to me.2011-08-19
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    I want to ask if there are other usual ways to interpret a matrix in vector space theory.2011-08-19
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    (2) "*Another way of stating the spectral theorem is to say that normal matrices are precisely those matrices that can be represented by a diagonal matrix with respect to a properly chosen orthonormal basis of $\mathbb{C}^n$. Phrased differently: a matrix is normal if and only if its eigenspaces span $\mathbb{C}^n$ and are pairwise orthogonal with respect to the standard inner product.*" - [Wikipedia](http://en.wikipedia.org/wiki/Normal_matrix#Consequences). (3) Inner products define angles, so positive definite maps are those which never take a vector $90^\circ$ or more from itself.2011-08-20
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    You might be interested in [this question](http://math.stackexchange.com/questions/9758/intuitive-explanation-of-a-positive-semidefinite-matrix).2011-08-20
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    @anon, Dylan: Thanks for clarifying a lot.2011-08-20

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