3
$\begingroup$

Suppose $A$ and $B$ are invertible $n \times n$ matrices that are similar to each other. Then for example, $A - 2I$ and $B - 2I$ are similar, and $A^{-1}$ and $B^{-1}$ are similar. What other operations will preserve similarity, and what algebraic object is the set of all invertible similar matrices (i.e. is it an algebra over some field, etc.)

  • 1
    Their transposes should be similar, too...2011-10-14
  • 2
    Let $f(x)$ be analytic, and defined $f(A)$ and $f(B)$ by their series expansions, then $f(A)$ and $f(B)$ are similar if $A$ and $B$ are...2011-10-14

2 Answers 2