352 CHAPTER 13. MATRICES AND THE INNER PRODUCT

Proof: First extend the given linearly independent set {w1, · · · ,wr} to a basis for Vand then apply the Gram Schmidt theorem to the resulting basis. Since {w1, · · · ,wr} isorthonormal it follows from Lemma 13.7.2 the result is of the desired form, an orthonormalbasis extending {w1, · · · ,wr}. ■

Recall Lemma 12.3.5 which is about preserving distances. It is restated here in the caseof an m×n matrix.

Lemma 13.7.4 Suppose R is an m×n matrix with m≥ n and R preserves distances. ThenR∗R = I.

With this preparation, here is the big theorem about the right polar factorization.

Theorem 13.7.5 Let F be an m×n matrix where m≥ n. Then there exists a Hermitian n×n matrix U which has all nonnegative eigenvalues and an m×n matrix R which preservesdistances and satisfies R∗R = I such that F = RU.

Proof: Consider F∗F. This is a Hermitian matrix because

(F∗F)∗ = F∗ (F∗)∗ = F∗F

Also the eigenvalues of the n×n matrix F∗F are all nonnegative. This is because if x is aneigenvalue,

λ (x,x) = (F∗Fx,x) = (Fx,Fx)≥ 0.

Therefore, by Lemma 13.7.1, there exists an n×n Hermitian matrix U having all nonneg-ative eigenvalues such that

U2 = F∗F.

Consider the subspace U (Fn). Let {Ux1, · · · ,Uxr} be an orthonormal basis for

U (Fn)⊆ Fn.

Note that U (Fn) might not be all of Fn. Using Lemma 13.7.3, extend to an orthonormalbasis for all of Fn,

{Ux1, · · · ,Uxr,yr+1, · · · ,yn} .

Next observe that {Fx1, · · · ,Fxr} is also an orthonormal set of vectors in Fm. This isbecause

(Fxk,Fx j) = (F∗Fxk,x j) =(U2xk,x j

)= (Uxk,U∗x j) = (Uxk,Ux j) = δ jk

Therefore, from Lemma 13.7.3 again, this orthonormal set of vectors can be extended to anorthonormal basis for Fm,

{Fx1, · · · ,Fxr,zr+1, · · · ,zm}

Thus there are at least as many zk as there are y j because m≥ n. Now for x ∈ Fn, since

{Ux1, · · · ,Uxr,yr+1, · · · ,yn}

is an orthonormal basis for Fn, there exist unique scalars,

c1 · · · ,cr,dr+1, · · · ,dn