354 CHAPTER 13. MATRICES AND THE INNER PRODUCT

13.8 The Square RootNow here is a uniqueness and existence theorem for the square root. It follows from thistheorem that U in the above right polar decomposition of Theorem 13.7.5 is unique.

Theorem 13.8.1 Let A be a self adjoint and nonnegative n× n matrix (all eigenvaluesare nonnegative). Then there exists a unique self adjoint nonnegative matrix B such thatB2 = A.

Proof: Suppose B2 = A where B is such a Hermitian square root for A with nonnegativeeigenvalues. Then by Theorem 13.1.6, B has an orthonormal basis for Fn of eigenvectors{u1, · · · ,un} .

Bui = µ iui

ThusB = ∑

iµ iuiu

∗i

because both linear transformations agree on the orthonormal basis. But this implies that

Aui = B2ui = µ2i ui

Thus these are also an orthonormal basis of eigenvectors for A. Hence, letting λ i = µ2i

A = ∑i

λ iuiu∗i , B = ∑

iλ

1/2i uiu

∗i

Let p(λ ) be a polynomial such that p(λ i) = λ1/2i . Say p(λ ) = a0 +a1λ · · ·+apλ

p. Then

Am =

(∑

iλ iuiu

∗i

)m

= ∑i1,··· ,im

λ i1ui1u∗i1λ i2ui2u

∗i2 · · ·λ imuimu

∗im (13.15)

= ∑i1,··· ,im

λ i1λ i2 · · ·λ imui1u∗i1ui2u

∗i2 · · ·uimu

∗im

= ∑i1,··· ,im

λ i1λ i2 · · ·λ imui1u∗imδ i1i2δ i2i3 · · ·δ im−1im

= ∑i1,··· ,im−1

λ i1λ i2 · · ·λ2im−1

ui1u∗im−1

δ i1i2δ i2i3 · · ·δ im−2im−1

...= ∑

i1

λmi1ui1u

∗i1 = ∑

iλ

mi uiu

∗i (13.16)

Therefore,p(A) = a0I +a1A · · ·+apAp

= a0 ∑iuiu

∗i +a1 ∑

iλ iuiu

∗i + · · ·+ap ∑

iλ

pi uiu

∗i

= ∑i

p(λ i)uiu∗i = ∑

iλ

1/2i uiu

∗i = B (13.17)