Exercise 9.9

Let $A$ be a $m\times n$ matrix, and $A=\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill \dots \hfill & \hfill a_n\hfill \end{array}\right]$, where $a_i$ are column vectors of dimension $m\times 1$

(a)

Since $$\parallel A{\text{∥}}_F^2=\sum _m\parallel {\text{row}}_m(A){\text{∥}}^2=\sum _n\parallel {\text{column}}_n(A){\text{∥}}^2$$

and

$$A{A}^T=\left[\begin{array}{ccc}\hfill a_1\hfill & \hfill \dots \hfill & \hfill a_n\hfill \end{array}\right]\left[\begin{array}{c}\hfill a_1^T\hfill \\ \hfill \dots \hfill \\ \hfill a_n^T\hfill \end{array}\right]=\sum _{i=1}^n{a}_i{a}_i^T$$

we have

$$\text{trace}(A{A}^T)=\text{trace}(\sum _{i=1}^n{a}_i{a}_i^T)=\sum _{i=1}^n\text{trace}(a_i{a}_i^T)=\sum _{i=1}^n\sum _{j=1}^m{a}_{\mathit{ij}}^2=\sum _{i=1}^n\parallel a_i{\text{∥}}^2=\parallel A{\text{∥}}_F^2.$$

Also

$$A^{T}A=\left[\begin{array}{cccc}\hfill a_1^T{a}_1\hfill & \hfill a_1^T{a}_2\hfill & \hfill \dots \hfill & \hfill a_1^T{a}_n\hfill \\ \hfill a_2^T{a}_1\hfill & \hfill a_2^T{a}_2\hfill & \hfill \dots \hfill & \hfill a_2^T{a}_n\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill a_n^T{a}_1\hfill & \hfill a_n^T{a}_2\hfill & \hfill \dots \hfill & \hfill a_n^T{a}_n\hfill \\ \hfill \hfill \end{array}\right]⟹\text{trace}(A^{T}A)=\sum _{i=1}^n{a}_i^T{a}_i=\sum _{i=1}^n\parallel a_i{\text{∥}}^2=\parallel A{\text{∥}}_F^2.$$

(b)

Apply the result from problem (a)

$$\begin{array}{llll}\hfill \parallel \mathit{UA}{V}^T{\text{∥}}_F^2& =\text{trace}({(\mathit{UA}{V}^T)}^T(\mathit{UA}{V}^T))& \hfill & \\ \hfill & =\text{trace}(V{A}^T{U}^T\mathit{UA}{V}^T)& \hfill & \\ \hfill & =\text{trace}(V{A}^{T}A{V}^T)& \hfill & \\ \hfill & =\text{trace}({(A{V}^T)}^T(A{V}^T))& \hfill & \\ \hfill & \text{apply }\mathit{trace}(X{X}^T)=\mathit{trace}(X^{T}X),X=A{V}^T& \hfill & \\ \hfill & =\text{trace}((A{V}^T){(A{V}^T)}^T)& \hfill & \\ \hfill & =\text{trace}((A{V}^T)(V{A}^T))& \hfill & \\ \hfill & =\text{trace}(A{A}^T)& \hfill & \\ \hfill & =\parallel A{\text{∥}}_F^2& \hfill & \\ \hfill & & \hfill \end{array}$$