362 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Proof: If F is a commuting family, it follows from Lemma 13.10.8 that it is simulta-neously diagonalizable. If it is simultaneously diagonalizable, then it follows from Lemma13.10.6 that it is a commuting family. ■

This is really a remarkable theorem. Recall that if S−1AS = D a diagonal matrix, thenthe columns of S are a basis of eigenvectors. Hence this says that when you have a com-muting family of non defective matrices, then they have the same eigenvectors. This showshow remarkable it is when a set of matrices commutes.

13.11 Fractional PowersThe main result is the following theorem.

Theorem 13.11.1 Let A be a self adjoint and nonnegative n× n matrix (all eigenvaluesare nonnegative) and let k be a positive integer. Then there exists a unique self adjointnonnegative matrix B such that Bk = A.

Proof: By Theorem 13.1.6, there exists an orthonormal basis of eigenvectors of A, say{vi}n

i=1 such that Avi = λ ivi with each λ i real. In particular, there exists a unitary matrix Usuch that

U∗AU = D, A =UDU∗

where D has nonnegative diagonal entries. Define B in the obvious way.

B≡UD1/kU∗

Then it is clear that B is self adjoint and nonnegative. Also it is clear that Bk = A. What ofuniqueness? Let p(t) be a polynomial whose graph contains the ordered pairs

(λ i,λ

1/ki

)where the λ i are the diagonal entries of D, the eigenvalues of A. Then

p(A) =UP(D)U∗ =UD1/kU∗ ≡ B

Suppose then that Ck = A and C is also self adjoint and nonnegative.

CB =Cp(A) =Cp(

Ck)= p

(Ck)

C = p(A)C = BC

and so {B,C} is a commuting family of non defective matrices. By Theorem 13.10.9 thisfamily of matrices is simultaneously diagonalizable. Hence there exists a single S such that

S−1BS = DB, S−1CS = DC

Where DC,DB denote diagonal matrices. Hence, raising to the power k, it follows that

A = Bk = SDkBS−1, A =Ck = SDk

CS−1

HenceSDk

BS−1 = SDkCS−1

and so DkB =Dk

C. Since the entries of the two diagonal matrices are nonnegative, this impliesDB = DC and so S−1BS = S−1CS which shows B =C. ■

A similar result holds for a general finite dimensional inner product space. See Problem21 in the exercises.