20220606 关于矩阵的Young不等式

定理：

合适维数的矩阵 $A,B,C,D$，合适维数的向量 $x,y$，有
$$2x^T\left(\left(AB\right)\otimes \left(CD\right)\right)y\le x^T\left(A{A}^T\otimes C{C}^T\right)x+y^T\left(B^{T}B\otimes D^{T}D\right)y$$

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原因分析：

$$\begin{array}{rl}& 2x^T\left(\left(AB\right)\otimes \left(CD\right)\right)y\\ =& 2x^T\left(A\otimes C\right)\left(B\otimes D\right)y\\ =& 2{\left({\left(A\otimes C\right)}^{T}x\right)}^T\left(B\otimes D\right)y\\ \le & {\left({\left(A\otimes C\right)}^{T}x\right)}^T\left({\left(A\otimes C\right)}^{T}x\right)+{\left(\left(B\otimes D\right)y\right)}^T\left(B\otimes D\right)y\\ =& x^T\left(A\otimes C\right){\left(A\otimes C\right)}^{T}x+y^T{\left(B\otimes D\right)}^T\left(B\otimes D\right)y\\ =& x^T\left(A{A}^T\otimes C{C}^T\right)x+y^T\left(B^{T}B\otimes D^{T}D\right)y\end{array}$$

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