Question:
If I have a matrix $A$, I know that its square root is a matrix that has the same eigenvectors as $A$ but its eigenvalues are the square roots of the eigenvalues of $A$.
What does this mean geometrically? If we think of $A$ as a transformation of space that takes a circle of unit vectors to a rotated ellipse, what does $\sqrt{A}$ do as a transformation?
Thoughts so far:
At first, I wanted to say $\sqrt{A}$ would give us the same rotated ellipse $A$ gives us, only that the lengths of the axes of that ellipse would be the square roots of the lengths of the axes of $A$.
However, that would only work if $A$ were a symmetric matrix and its orthonormal eigenvectors are the axes of the ellipse. If $A$ isn't symmetric...