Interpretation of matrices in vector spaces 
*

*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?

*I was wondering how to interpret a normal matrix (i.e. a square
matrix $A$ s.t. $A^* A=AA^*$) in vector spaces?

*What kind of linear mappings is represented as a positive definite
matrix under some possibly special basis?


Thanks and regards!
 A: *

*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. 

*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.

*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. 
