在数学中,弗罗比尼乌斯内积(英語:Frobenius inner product)是一种基于两个矩阵的二元运算,结果是一个数值。它常常被记为
。这个运算是一個將矩陣視為向量的逐元素内积。参与运算的两个矩阵必须有相同的维度、行数和列数,但不局限于方阵。
定义
给定两个n×m维複矩阵 A和B:
![{\displaystyle \mathbf {A} ={\begin{pmatrix}A_{11}&A_{12}&\cdots &A_{1m}\\A_{21}&A_{22}&\cdots &A_{2m}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nm}\\\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}B_{11}&B_{12}&\cdots &B_{1m}\\B_{21}&B_{22}&\cdots &B_{2m}\\\vdots &\vdots &\ddots &\vdots \\B_{n1}&B_{n2}&\cdots &B_{nm}\\\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fcd37e456ba99e37d2398d56ead3992d39c3365)
弗罗比尼乌斯内积定义为如下的矩阵元素求和
![{\displaystyle \langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }=\sum _{i,j}{\overline {A_{ij}}}B_{ij}=\mathrm {tr} \left({\overline {\mathbf {A} ^{T}}}\mathbf {B} \right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b38d9e39f4d796fa5523ae6945c7233b1d9440d2)
其中上划线表示复数和複矩阵的共轭操作。若將定義詳細寫出,則有
![{\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }=&{\overline {A}}_{11}B_{11}+{\overline {A}}_{12}B_{12}+\cdots +{\overline {A}}_{1m}B_{1m}\\&+{\overline {A}}_{21}B_{21}+{\overline {A}}_{22}B_{22}+\cdots +{\overline {A}}_{2m}B_{2m}\\&\vdots \\&+{\overline {A}}_{n1}B_{n1}+{\overline {A}}_{n2}B_{n2}+\cdots +{\overline {A}}_{nm}B_{nm}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/328e9667ee15a37482e7c3b3f5b4647e0913a0bb)
此計算與點積十分相似,所以是一個內積的範例。
性质
弗罗比尼乌斯内积是半双线性形式。给定複矩阵A, B, C, D, 以及复数a和b,我们有
![{\displaystyle \langle a\mathbf {A} ,b\mathbf {B} \rangle _{\mathrm {F} }={\overline {a}}b\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffa78f569f7441402a71ac20db9d21ea5e2672eb)
![{\displaystyle \langle \mathbf {A} +\mathbf {C} ,\mathbf {B} +\mathbf {D} \rangle _{\mathrm {F} }=\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }+\langle \mathbf {A} ,\mathbf {D} \rangle _{\mathrm {F} }+\langle \mathbf {C} ,\mathbf {B} \rangle _{\mathrm {F} }+\langle \mathbf {C} ,\mathbf {D} \rangle _{\mathrm {F} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8f4a5af0344c3a3b919d0276608c42164e5bb6c)
并且,交换複矩阵的次序所得到的是原来结果的共轭矩阵:
![{\displaystyle \langle \mathbf {B} ,\mathbf {A} \rangle _{\mathrm {F} }={\overline {\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/802dace16b884dc7e5b5199a0590a345f9f7bcc3)
对于相同的矩阵,有
![{\displaystyle \langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }\geq 0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a3fd5503f1b588d2496f34cd751e07553ac561)
样例
实矩阵
给定实矩阵:
![{\displaystyle \mathbf {A} ={\begin{pmatrix}2&0&6\\1&-1&2\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}8&-3&2\\4&1&-5\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d02c4fa2189b2b5087d64731a3b37cd6622c6419)
则:
![{\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }&=2\cdot 8+0\cdot (-3)+6\cdot 2+1\cdot 4+(-1)\cdot 1+2\cdot (-5)\\&=16+12+4-1-10\\&=21\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49e2b589b2782b133a01d4608dd7c6c88647f7f5)
复矩阵
给定复矩阵
![{\displaystyle \mathbf {A} ={\begin{pmatrix}1+i&-2i\\3&-5\end{pmatrix}}\,,\quad \mathbf {B} ={\begin{pmatrix}-2&3i\\4-3i&6\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e068d3fda95e0a59d64a96e30b7a8a9efb51ea)
那么它们的共轭 (非转置) 矩阵为
![{\displaystyle {\overline {\mathbf {A} }}={\begin{pmatrix}1-i&+2i\\3&-5\end{pmatrix}}\,,\quad {\overline {\mathbf {B} }}={\begin{pmatrix}-2&-3i\\4+3i&6\end{pmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebda52411a71d947ff29299e83bcbe08e6fabdcc)
因此,
![{\displaystyle {\begin{aligned}\langle \mathbf {A} ,\mathbf {B} \rangle _{\mathrm {F} }&=(1-i)\cdot (-2)+(+2i)\cdot 3i+3\cdot (4-3i)+(-5)\cdot 6\\&=(-2+2i)+-6+12-9i+-30\\&=-26-7i\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d76db8725d3f2d4479730c6102b7214822ebb977)
但注意
![{\displaystyle {\begin{aligned}\langle \mathbf {B} ,\mathbf {A} \rangle _{\mathrm {F} }&=(-2)\cdot (1+i)+(-3i)\cdot (-2i)+(4+3i)\cdot 3+6\cdot (-5)\\&=-26+7i\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80310ae28114893de81d8da9d7ff75be0ec70b84)
A、B与其本身的弗罗比尼乌斯内积分别为
![{\displaystyle \langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }=2+4+9+25=40}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d841d263f5b5ed473d6207c7e7dce42513b1bf31)
![{\displaystyle \langle \mathbf {B} ,\mathbf {B} \rangle _{\mathrm {F} }=4+9+25+36=74}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28214e04ae136ad57e38ecbe310f63784c2fa370)
弗罗比尼乌斯范数
从弗罗比尼乌斯内积我们可以诱导出弗罗比尼乌斯范数
![{\displaystyle \|\mathbf {A} \|_{\mathrm {F} }={\sqrt {\langle \mathbf {A} ,\mathbf {A} \rangle _{\mathrm {F} }}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31ba35f7f9ba636be19c11bffb0938a67b739a41)
參考資料
相關條目