12.6 Fractional Powers
The main result is the following theorem.
Theorem 12.6.1 Let A be a self adjoint and nonnegative n × n matrix (all eigenvalues
are nonnegative) and let k be a positive integer. Then there exists a unique self adjoint
nonnegative matrix B such that Bk = A.
Proof: By Theorem 12.3.2 or Corollary 6.4.12, there exists an orthonormal basis of
eigenvectors of A, say
such that Avi
with each λi
real. In particular, there exists
a unitary matrix U
where D has nonnegative diagonal entries. Define B in the obvious way.
Then it is clear that B is self adjoint and nonnegative. Also it is clear that Bk = A. What of
uniqueness? Let p
be a polynomial whose graph contains the ordered pairs
are the diagonal entries of D,
the eigenvalues of A
Suppose then that Ck = A and C is also self adjoint and nonnegative.
is a commuting family of non defective matrices. By Theorem
family of matrices is simultaneously diagonalizable. Hence there exists a single S
Where DC,DB denote diagonal matrices. Hence, raising to the power k, it follows
and so DBk = DCk. Since the entries of the two diagonal matrices are nonnegative, this implies
DB = DC and so S−1BS = S−1CS which shows B = C. ■
A similar result holds for a general finite dimensional inner product space. See Problem 22 in