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Basics

0.1 Basic Content $$ \begin{align} (\boldsymbol{AB})^{-1} &= \boldsymbol{B}^{-1}\boldsymbol{A}^{-1} \cr (\boldsymbol{ABC\cdots})^{-1} &= \cdots\boldsymbol{C}^{-1}\boldsymbol{B}^{-1}\boldsymbol{A}^{-1} \cr (\boldsymbol{A}^\top)^{-1} &= (A^{-1})^\top \cr (\boldsymbol{A} + \boldsymbol{B})^\top &= \boldsymbol{A}^\top + \boldsymbol{B}^\top \cr (\boldsymbol{AB})^\top &= \boldsymbol{B}^\top\boldsymbol{A}^\top \cr (\boldsymbol{ABC\cdots})^\top &= \cdots\boldsymbol{C}^\top\boldsymbol{B}^\top\boldsymbol{A}^\top \cr (\boldsymbol{A}^{H})^{-1} &= (\boldsymbol{A}^{-1})^{H} \cr (\boldsymbol{A} + \boldsymbol{B})^H &= \boldsymbol{A}^H + \boldsymbol{B}^H \cr (\boldsymbol{AB})^H &= \boldsymbol{B}^H\boldsymbol{A}^H \cr (\boldsymbol{ABC\cdots})^H &= \cdots\boldsymbol{C}^H\boldsymbol{B}^H\boldsymbol{A}^H \end{align} $$ 0.2 Trace $$ \begin{align} \text{TR}(\boldsymbol{A}) &= \sum_{i}A_{ii} \cr \text{TR}(\boldsymbol{A}) &= \sum_{i}\lambda_{i}, \quad \lambda_{i}=\text{eig}(\boldsymbol{A})_{i} \cr \text{TR}(\boldsymbol{A}) &= \text{TR}(\boldsymbol{A}^\top) \cr \text{TR}(\boldsymbol{AB}) &= \text{TR}(\boldsymbol{BA}) \cr \text{TR}(\boldsymbol{A+B}) &= \text{TR}(\boldsymbol{A}) + \text{TR}(\boldsymbol{B}) \cr \text{TR}(\boldsymbol{ABC}) &= \text{TR}(\boldsymbol{BCA}) = \text{TR}(\boldsymbol{CAB}) \cr \boldsymbol{a}^\top\boldsymbol{a} &= \text{Tr}(\boldsymbol{aa}^\top) \end{align} $$

• Linear Algebra
• Matrix

Monday, November 25, 2024 | 2 minutes Read

Notations

With reference to The Matrix Cookbook 0.1 Notation and Nomenclature Notations $\boldsymbol{A}$ $\boldsymbol{A}_{ij}$ $\boldsymbol{A}_i$ $\boldsymbol{A}^{ij}$ $\boldsymbol{A}^{n}$ $\boldsymbol{A}^{-1}$ $\boldsymbol{A}^{+}$ $\boldsymbol{A}^{1/2}$ $(\boldsymbol{A})_{ij}$ $\boldsymbol{A}_{ij}$ $[\boldsymbol{A}]_{ij}$ $\boldsymbol{a}$ $\boldsymbol{a}_i$ $a_i$ $a$ $\mathfrak{R}z$ $\mathfrak{R}\boldsymbol{z}$ $\mathfrak{R}\boldsymbol{Z}$ $\mathfrak{F}z$ $\mathfrak{F}\boldsymbol{z}$ $\mathfrak{F}\boldsymbol{Z}$ $\det(\boldsymbol{A})$ $\text{Tr}(\boldsymbol{A})$ $\text{diag}(\boldsymbol{A})$ $\text{eig}(\boldsymbol{A})$ $\text{vec}(\boldsymbol{A})$ $\text{sup}$ $||\boldsymbol{A}||$ $\boldsymbol{A}^\top$ $\boldsymbol{A}^{-\top}$ $\boldsymbol{A}^{*}$ $\boldsymbol{A}^H$ $\boldsymbol{A}\circ\boldsymbol{B}$ $\boldsymbol{A}\otimes\boldsymbol{B}$ $\boldsymbol{0}$ $\boldsymbol{I}$ $\boldsymbol{J}^{ij}$ $\boldsymbol{\Sigma}$ $\boldsymbol{\Lambda}$ Nomenclature Matrix Matrix indexed for some purpose Matrix indexed for some purpose Matrix indexed for some purpose

• Linear Algebra
• Matrix

Monday, November 25, 2024 | 2 minutes Read