Linear transformations on matrices: the invariance of rank k matrices

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...

Matrices and Linear Transformations

Commuting maps on rank-k matrices

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 ...

On higher rank numerical hulls of normal matrices

‎In this paper‎, ‎some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated‎. ‎A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given‎. ‎Moreover‎, ‎using the extreme points of the numerical range‎, ‎the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$‎, ‎where $A_1...