Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name? - MathOverflow

Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2009-10-12T01:13:49Z

This is also the space of real, symmetric bilinear forms in R^n.

Answer by Scott Morrison for Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2009-10-12T01:45:24Z

For starters, since they're real I'd say symmetric instead of self-adjoint.

Answer by Darsh Ranjan for Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2009-10-14T09:09:51Z

Two possible answers:

Standard jargon is SPD (for "symmetric positive-definite").
This isn't exactly a "name," but the n x n symmetric positive-definite matrices are exactly those matrices A such that the bilinear function (x, y) -> y^TAx defines an inner product on Rⁿ. Conversely, every bilinear function is of that form for some A, so with some abuse of terminology, you could equate the set of those matrices with the set of inner products on Rⁿ.

There are many other ways to characterize SPD matrices, but that's the only one I can think of at the moment that can be summarized as a single noun phrase.

Answer by Lior Silberman for Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2009-10-14T18:38:59Z

This is the symmetric space of GL_n(R)

Answer by Mark Meckes for Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2009-10-23T19:15:45Z

Note that this space is not a vector space, but is a convex cone in the vector space of nxn matrices (it is closed under addition and multiplication by positive scalars). Hence people sometimes refer to the "positive semidefinite cone".

Answer by Jonas Meyer for Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2009-11-05T09:10:54Z

How about ? I have seen or used to denote the set of positive linear transformations in a set of linear transformations on an inner product space, but this was in the context of operator algebras.

Answer by Kjetil B Halvorsen for Does the space of n x n, positive-definite, self-adjoint, real matrices have a better name?

2012-05-25T19:05:45Z

It is often usefull to know that this set can be identified with the set of non-singulat covariance matrices of random vectors with values in $\mathbb(R)^n$.