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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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    @anon, Dylan: Thanks for clarifying a lot.2011-08-20

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  1. A matrix always represents a linear transformation between two vector spaces, and every other use of a matrix in linear algebra is a special case of this. A bilinear form, being a bilinear map $V \times V \to k$, is the same as a linear map $V \to V^{\ast}$, for example.

  2. Any matrix which is diagonalizable with orthogonal eigenvectors is necessarily normal. The spectral theorem says that the converse is true. Abstractly, the spectral theorem can be thought of as a statement about commutative $C^{\ast}$-algebras generated by an operator, especially in light of the Gelfand representation. This is the point of view that generalizes best to the infinite-dimensional situation.

  3. Positive-definiteness is properly thought of as a property of a bilinear form $V \times V \to \mathbb{R}$ (or a sesquilinear form, but let's ignore this case for now). It is possible to represent such a thing with a matrix since it is the same thing as a linear map $V \to V^{\ast}$, but note that these are not the same vector space.

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    @Tim: 1. they are the same, if you equip $V$ with a basis $e_1, ... e_n$ and $V^{\ast}$ with the dual basis $e_1^{\ast}, ... e_n^{\ast}$. 2. Positive definite matrices should not be thought of as representing linear mappings, if by this you mean maps $V \to V$. They describe bilinear forms.2011-08-20