Kenneth Kuttler

92 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRA

Corollary 5.8.9 If A is a real symmetric matrix, then A is Hermitian and there exists a realunitary matrix, U such that UT AU = D where D is a diagonal matrix.

Proof: This follows from Theorem 5.8.4 and Corollary 5.8.8.

5.9 The Right Polar DecompositionThe right polar decomposition involves writing a matrix as a product of two other matrices,one which preserves distances and the other which stretches and distorts. First here aresome lemmas.

Lemma 5.9.1 Let A be a Hermitian matrix such that all its eigenvalues are nonnegative.Then there exists a Hermitian matrix, A1/2 such that A1/2 has all nonnegative eigenvaluesand

(A1/2

)2= A.

Proof: Since A is Hermitian, there exists a diagonal matrix D having all real non-negative entries and a unitary matrix U such that A = U∗DU. Then denote by D1/2 thematrix which is obtained by replacing each diagonal entry of D with its square root. ThusD1/2D1/2 = D. Then define

A1/2 ≡U∗D1/2U.

Then (A1/2

)2=U∗D1/2UU∗D1/2U =U∗DU = A.

Since D1/2 is real, (U∗D1/2U

)∗=U∗

(D1/2

)∗(U∗)∗ =U∗D1/2U

so A1/2 is Hermitian. This proves the lemma.There is also a useful observation about orthonormal sets of vectors which is stated in

the next lemma.

Lemma 5.9.2 Suppose {x1,x2, · · · ,xr} is an orthonormal set of vectors. Then if c1, · · · ,crare scalars, ∣∣∣∣∣ r

∑k=1

ckxk

∣∣∣∣∣2

∑k=1|ck|2 .

Proof: This follows from the definition. From the properties of the dot product andusing the fact that the given set of vectors is orthonormal,∣∣∣∣∣ r

∑k=1

ckxk

∣∣∣∣∣2

∑k=1

ckxk,r

∑j=1

c jx j

)

= ∑k, j

ckc j (xk,x j) =r

∑k=1|ck|2 .

This proves the lemma.Next it is helpful to recall the Gram Schmidt algorithm and observe a certain property

stated in the next lemma.

92 CHAPTER 5. SOME IMPORTANT LINEAR ALGEBRACorollary 5.8.9 [fA is a real symmetric matrix, then A is Hermitian and there exists a realunitary matrix, U such that U' AU = D where D is a diagonal matrix.Proof: This follows from Theorem 5.8.4 and Corollary 5.8.8.5.9 The Right Polar DecompositionThe right polar decomposition involves writing a matrix as a product of two other matrices,one which preserves distances and the other which stretches and distorts. First here aresome lemmas.Lemma 5.9.1 Let A be a Hermitian matrix such that all its eigenvalues are nonnegative.Then there exists a Hermitian matrix, A'/? such that A‘/? has all nonnegative eigenvaluesand (at/2)? =A.Proof: Since A is Hermitian, there exists a diagonal matrix D having all real non-negative entries and a unitary matrix U such that A = U*DU. Then denote by D!/? thematrix which is obtained by replacing each diagonal entry of D with its square root. ThusD'/2p!/? = D. Then defineAl? =u*D!/U.Then 5(a'?) =u*D'?uu*D'?u =U*DU =A.Since D!/? is real,(u'p'?u) =U* (p'”) (U*)* _ u*pD'/uUso A!/? is Hermitian. This proves the lemma.There is also a useful observation about orthonormal sets of vectors which is stated inthe next lemma.Lemma 5.9.2 Suppose {x,,X2,--: ,X;} is an orthonormal set of vectors. Then if c,,-+++ ,Crare scalars,2r r¥ cexe| = ¥ lexl’-k=1 k=1Proof: This follows from the definition. From the properties of the dot product andusing the fact that the given set of vectors is orthonormal,r r rY? Cex = e an Bai]k=1 k=1 J=12r= PV ceej (xx) = Y lel.kj k=1This proves the lemma.Next it is helpful to recall the Gram Schmidt algorithm and observe a certain propertystated in the next lemma.