Linear transformations that map the P-matrices into themselves

Matrices and Linear Transformations

Linear Transformations on Matrices *

Even in thi s generality , it is clear that .!l' (I , ~) is a multiplicative se migroup with an ide ntity. The invariant I can be a scalar valued fun ction, e.g. , I (X) = det (X) ; or for that matter it can describe a property, e.g., m can equal M" (C) and I (X) can mean that X is unitary , so that we are simply asking for the s tructure of all linear transformation s T that map the unitary gr...

Linear Map of Matrices

The paper is concerned with a generalization of concepts introduced in [13], i.e. introduced are matrices of linear transformations over a finitedimensional vector space. Introduced are linear transformations over a finitedimensional vector space depending on a given matrix of the transformation. Finally, I prove that the rank of linear transformations over a finite-dimensional vector space is ...

2: Linear Transformations and Matrices

The general approach to the foundations of mathematics is to study certain spaces, and then to study functions between these spaces. In this course we follow this paradigm. Up until now, we have been studying properties of vector spaces. Vector spaces have a linear structure, and so it is natural to deal with functions between vector spaces that preserve this linear structure. That is, we will ...

Things that can be made into themselves

One says that a property P of sets of natural numbers can be made into itself iff there is a numbering α0, α1, . . . of all left-r.e. sets such that the index set {e : αe satisfies P} has the property P as well. For example, the property of being Martin-Löf random can be made into itself. Herein we characterize those singleton properties which can be made into themselves. A second direction of ...