Proving existence and uniqueness of a matrix,

Question

Let A be nxn with real coefficients and assume that it has n distinct eigenvalues, and all eigenvalues are positive real numbers.

Let k $\ge$3 be an odd integer.

a) Prove there exists a unique real matrix B with $B^k$ = A.

b) How many complex matrices B satisfy $B^k$ = A. (Include the real matrices B in your count.)

I would like hints only.

I know, from my previous questions on MSE, that positive real eigenvalues does not imply A is symmetric, and hence not necessarily positive-definite.

I also know that A is diagonalizable (not necessarily orthogonally diagonalizable), because of the n distinct eigenvalues it has - so $$A = SDS^{-1}$$,

where S's columns are eigenvectors of A.

...now I need to somehow make use of the other information given, namely that the eigenvalues are positive and real.

Thanks,

Answer 1

Hint: $B$ is diagonalized by the same $S$.

Answer 2

Hints:

• $B$ is diagonalized by the same $S$.
• The entries of $D$ correspond to the eigenvalues of $A$.