Problem on singular value and trace of matrix Let $A,B\in \mathbb{R}^{n\times n}$ show that there exists 
(1) Orthogonal matrix U,V satisfying  $|trace(AB)|\leq\sum_{i=1}^{n}\sum_{j=1}^{n}(\sigma_{i}(A)\sigma_{j}(B)|u_{ij}v_{ij}|)$
(2) Double stochastic matrix $C$ satisfying
$|trace(AB)|\leq\sum_{i=1}^{n}\sum_{j=1}^{n}(\sigma_{i}(A)\sigma_{j}(B)c_{ij})\leq\sum_{i=1}^{n}(\sigma_{i}(A)\sigma_{i}(B))$
Note that $\sigma_{1}(M)\geq ...\sigma_{n}(M)\geq 0$ are ordered singular value of matrix $M$
I do know the definition of singular value and orthogonal matrix but only a few properties about them. Maybe singular value decomposition can deal with the problems but I had trouble expressing its product. Any ideas how to find these matrixes and what properties or theorems should I apply here?
Thanks for your help.
 A: Hint for (1): $A=\sum_{i=1}^n \sigma_iu_iv_i^T$ where the vectors $u_i,v_i$ correspond to the SVD. If you have $A,B$, then they both have a corresponding SVD with $u_i',v_i'$ so write it out and expand the product of $AB$ as a double sum, giving terms like $\sigma_i(A)\sigma_j(B)u_iv_i^T u'_jv_j'^T$. Now notice that $U:=u_iv^T_i$ and $V:=u_j'v'^T_j$ are unitary.
A: Definitions and properties
A orthogonal matrix $\mathbf{U}\in\mathbb{R}^{n\times n}$ matrix satisfies
$$
 \mathbf{U}^{\mathrm{T}} \mathbf{U} = \mathbf{U}\, \mathbf{U}^{\mathrm{T}}= \mathbf{I}_{n}.
$$
Such a matrix has full rank, and all eigenvalue are $\pm 1$, which implies the determinant:
$$
  \det \mathbf{U} = \pm 1.
$$
The singular value decomposition of a rank $\rho$ matrix $\mathbf{A}\in\mathbb{R}^{n\times n}_{\rho}$ is
$$
  \mathbf{A} = \mathbf{U} \, \Sigma \, \mathbf{V}^{*} =
%
\left[ \begin{array}{cc}
  \color{blue}{\mathbf{U}_{\mathcal{R}\left(\mathbf{A}\right)}} &
  \color{red} {\mathbf{U}_{\mathcal{N}\left(\mathbf{A}^{*}\right)}}
\end{array} \right]
%
\Sigma 
%
\left[ \begin{array}{c}
  \color{blue}{\mathbf{V}_{\mathcal{R}\left(\mathbf{A}^{*}\right)}}^{*} \\
  \color{red} {\mathbf{V}_{\mathcal{N}\left(\mathbf{A}\right)}}^{*}
\end{array} \right]
%
$$
The coloring identifies entities in $\color{blue}{range}$ and $\color{red}{null}$ spaces.
The ordered singular values
$$
  \sigma_{1} \ge \sigma_{2} \ge \dots \ge \sigma_{\rho} > 0
$$
form the diagonal matrix
$$
  \mathbf{S}_{k,k} = \sigma_{k}, \quad k = 1, \rho.
$$
Embed this in a sabot matrix
$$
  \Sigma = \left[ \begin{array}{cc} 
  \mathbf{S} & \mathbf{0} \\ \mathbf{0} & \mathbf{0}
\end{array} \right]
$$
The domain matrices are orthonormal spans. For the range spaces:
$$
\begin{align}
 \color{blue}{\mathcal{R}\left( \mathbf{A} \right)} 
&= \text{span} 
\left\{
  \color{blue}{u_{1}}, \color{blue}{u_{2}}, \dots, \color{blue}{u_{\rho}}
\right\} ,\\
%
 \color{blue}{\mathcal{R}\left( \mathbf{A}^{*} \right)} 
&= \text{span} 
\left\{
  \color{blue}{v_{1}}, \color{blue}{v_{2}}, \dots, \color{blue}{v_{\rho}}
\right\} .
\end{align}
$$
For the null spaces:
$$
\begin{align}
 \color{red}{\mathcal{N}\left( \mathbf{A} \right)} 
&= \text{span} 
\left\{
  \color{red}{u_{\rho+1}}, \color{red}{\rho+u_{2}}, \dots, \color{red}{u_{n}}
\right\} ,\\
%
 \color{red}{\mathcal{N}\left( \mathbf{A}^{*} \right)} 
&= \text{span} 
\left\{
  \color{red}{v_{\rho+1}}, \color{red}{v_{\rho+2}}, \dots, \color{red}{v_{n}}
\right\} .
\end{align}
$$
Tools
The SVD resolves the matrix $\mathbf{A}$ into $\rho$ mutually orthogonal components:
$$
  \mathbf{A} = \sum_{k=1}^{\rho} \sigma_{k} 
  \color{blue}{u_{k}} 
  \color{blue}{v_{k}^{\mathrm{T}}}
$$
$$
 \text{tr }\left( \mathbf{A} \mathbf{B} \right) =
%
 \sum_{k=1}^{n} \left| \mathbf{A} \mathbf{B} \right|_{k,k} =
%
 \sum_{k=1}^{n} \mathbf{A}_{k,:} \mathbf{B}_{:,k} =
%
 \sum_{j=1}^{n} \sum_{k=1}^{n} a_{j,k} b_{k,j}
$$
