91Ó°ÊÓ

(a)

Let$A=U\underset{}{\overset{}{∑}}{V}^T$ be the singular value decomposition of A . Rewrite A as follows.

$A=U\underset{}{\overset{}{∑}}{V}^T=U{V}^{T}V\underset{}{\overset{}{∑}}{V}^T=\left(U{V}^T\right)\left(U\underset{}{\overset{}{∑}}{V}^T\right).......\left(1\right)$

Define $Q=U{V}^T$and $S=V\underset{}{\overset{}{∑}}{V}^T$.

The columns of V are orthogonal by construction. If $v_i$is the $i^{th}$column of V , then $V^{T}V=v_i^T{v}_i=I$. The columns of U also are orthogonal. Hence, by the same argument,

$U^{T}U=I=U{U}^T$.

This gives the following.

$Q{Q}^T=U{V}^{T}U{V}^T=U{V}^{T}V{U}^T=UI{U}^T=I$

This proves that is Q orthogonal.

Let's check if is symmetric.

role="math" localid="1660737512505" $S^T=V\underset{}{\overset{}{∑}}{V}^T=V\underset{}{\overset{T}{∑}}{V}^T=V\underset{}{\overset{}{∑}}{V}^T=S$

$\underset{}{\overset{T}{∑}}=\underset{}{\overset{}{∑}}$, since$\underset{}{\overset{}{∑}}$ is a diagonal matrix.

Now, let's check if S is positive semi-definite.

role="math" localid="1660737787413" $x^{T}Sx=x^{T}V\underset{}{\overset{}{∑}}{V}^{T}x=x^{T}V\underset{}{\overset{}{∑}}{V}^{T}x={\left(V^{T}x\right)}^T\underset{}{\overset{}{∑}}\left(V^{T}x\right)=y^T\underset{}{\overset{}{∑}}yâ‰\yen 0$

$y^T\underset{}{\overset{}{∑}}yâ‰\yen 0$for all $y$since since $\underset{}{\overset{}{∑}}$is a positive semi-definite as all of its eigenvalues are non-negative.

This proves that $A=QS$where Q is orthogonal and S is symmetric and positive semi-definite.

(b)

Rewrite A as follows.

$A=U\underset{}{\overset{}{∑}}{V}^T=U\underset{}{\overset{}{∑}}U{V}^T=U\underset{}{\overset{}{∑}}{U}^{T}U{V}^T=S_1{Q}_1$

By the arguments similar to the ones in part (a),role="math" localid="1660738317346" $S_1=U\underset{}{\overset{}{∑}}{U}^T$ is symmetric and positive semi-definite and $Q_1=U{V}^T$is orthogonal.

a. Write $A=U\underset{}{\overset{}{∑}}{V}^T=U{V}^{T}V\underset{}{\overset{}{∑}}{V}^T=\left(U{V}^T\right)\left(V\underset{}{\overset{}{∑}}{V}^T\right)=QS$.Show that Q is orthogonal and S is symmetric positive semi-definite.

b. Write $A=U\underset{}{\overset{}{∑}}{V}^T=U\underset{}{\overset{}{∑}}{U}^{T}U{V}^T=\left(U\underset{}{\overset{}{∑}}{U}^T\right)\left(U{V}^T\right)=S_1{Q}_1$. Show that $Q_1$is orthogonal and $S_1$is symmetric positive semi-definite.