F
范数:设矩 \(\mathbf A=(a_{i,j})_{m\times n}\) ,则其F
范数为 \(||\mathbf A||_F=\sqrt{\sum_{i,j}a_{i,j}^{2}}\) 。
它是向量 \(L_2\) 范数的推广。F
范数等 \(\mathbf A\mathbf A^T\) 的迹的平方根 \(||\mathbf A||_F=\sqrt{tr(\mathbf A \mathbf A^{T})}\) 。Hadamard product
(又称作逐元素积):
\[\mathbf A \circ \mathbf B =\begin{bmatrix} a_{1,1}b_{1,1}&a_{1,2}b_{1,2}&\cdots&a_{1,n}b_{1,n}\\ a_{2,1}b_{2,1}&a_{2,2}b_{2,2}&\cdots&a_{2,n}b_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}b_{m,1}&a_{m,2}b_{m,2}&\cdots&a_{m,n}b_{m,n}\end{bmatrix}\]Kronnecker product
:
\[\mathbf A \otimes \mathbf B =\begin{bmatrix}a_{1,1}\mathbf B&a_{1,2}\mathbf B&\cdots&a_{1,n}\mathbf B\\ a_{2,1}\mathbf B&a_{2,2}\mathbf B&\cdots&a_{2,n}\mathbf B\\ \vdots&\vdots&\ddots&\vdots\\ a_{m,1}\mathbf B&a_{m,2}\mathbf B&\cdots&a_{m,n}\mathbf B \end{bmatrix}\]