Geodesic
Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 739-748

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Let Mm (F) be the vector space of m-square matrices X — (Xij), i,j= 1, ... , m over a field ƒ;ƒ a function on Mm (F) to some set R. It is of interest to determine the structure of the linear maps T: Mm (F) → Mm(F) that preserve the values of the function ƒ(i.e., ƒ(T(x)) — ƒ(x) for all X). For example, if we take ƒ(x) to be the rank of X, we are asking for a determination of the types of linear operations on matrices that preserve rank (6). Other classical invariants that may be taken for ƒ are the determinant, the set of eigenvalues, and the rth elementary symmetric function of the eigenvalues. 
Botta, Peter. Linear Transformation on Matrices: The Invariance of a Class of General Matrix Functions. II. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 739-748. doi: 10.4153/CJM-1968-073-5
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