Let be an invertible matrix. Show that there exists a unique differentiable function such that
If is invertible, then there exists some diagonal matrix and similar matrix such that
Since the diagonal matrix is composed of eigenvalues along its main diagonal,
There are two important properties of square matrices: for some square matrix , the sum of its eigenvalues equals its trace; the product of its eigenvalues equals its determinant [Proof].
Hence, we need to find a function such that
Only the exponential function satisfies this criteria.
Check out my other notes at Proof, Disproof, and Derivation