Throughout the following, we base everything on the matrix $\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$and explicitly denote the dimensions. If the definition is valid for an arbitrary [[Field]], we denote it by $F$, otherwise we explicitly use e.g. $\mathbb{R}$ or $\mathbb{C}$. >[!note] >- For derivatives involving matrices, see **[[Matrix Calculus]]**. >- For the [[Group|group]] structure resulting from invertible matrices, see **[[General Linear Group and Matrix Groups]]**. >[!brainwaves] In pure Mathematics >Matrices represent [[Linear Maps and Operators]] --- - [[Determinant of a Square Matrix]] - [[Kernel or Nullspace of a linear Map]] - [[Rank of a Matrix]] - [[Trace of a Square Matrix]] - [[Definiteness]] --- #### Special Maps of a Matrix $\mathbf{A}$ | **Special Matrices** | **Definition** | **Properties** | **Remarks** | | -------------------- | -------------------------------------------- | ---------------------------------------------------------------------- | ------------------------------------------------ | | **Transpose** | $[\mathbf{A}^{T}]_{ij}=[\mathbf{A}]_{ij}$ | $(\mathbf{A+B})^{T}=\mathbf{A}^{T}+\mathbf{B}^{T}$ | | | | | $(\mathbf{AB})^{T}=\mathbf{B}^{T}\mathbf{A}^{T}$ | | | | | $(\mathbf{A}^{T})^{-1}=(\mathbf{A}^{-1})^{T}$ | | | | | $\text{det}(\mathbf{A}^{T})=\text{det}(\mathbf{A})$ | | | **Inverse** | $\mathbf{A}\cdot \mathbf{A}^{-1}=\mathbf{I}$ | $(\mathbf{A \cdot B})^{-1}=\mathbf{B}^{-1}\cdot \mathbf{A}^{-1}$ | Only defined for square matrices | | | | $(c\mathbf{A})^{-1}=c^{-1}\mathbf{A}^{-1}$ | | | | | $\det(\mathbf{A}^{-1})=(\det(\mathbf{A}))^{-1}$ | | | **Projector** | $\mathbf{A}\cdot \mathbf{A}=\mathbf{A}$ | othogonal, if $\langle \mathbf{x}-\mathbf{Ax}, \mathbf{Ax} \rangle =0$ | Orthogonal, if projector matrix symmetric | | | | | Only defined for square matrices | --- #### Special Matrix Structures - **Symmetric** $\mathbf{A}^{T}=\mathbf{A}$ - Sum and difference of symmetric matrices is symmetric, product not - If $\mathbf{A}$ symmetric and $n \in \mathbb{N}$, $\mathbf{A}^{n}$ is symmetric - If $\mathbf{A}^{-1}$ exists, it is symmetric if and only if $\mathbf{A}$ is symmetric - **Skew-Symmetric** $\mathbf{A}^\top=-\mathbf{A}$ - Sum and scalar multiples of skew symmetric matrices are skew symmetric - trace, determinant $0$ - - **Hermitian** $\mathbf{A}=\mathbf{A}^{H}=\overline{\mathbf{A}^{T}}$ - Complex conjugate of every element - $\langle \mathbf{w}, \mathbf{Av}\rangle=\langle \mathbf{v}, \mathbf{Aw}\rangle$ for any pair of vector - For $\mathbf{v}\in \mathbb{C}^{n}$ it holds that $\langle \mathbf{v}, \mathbf{Av}\rangle\ \in \mathbb{R}$ - $(\mathbf{AB})^{H}=\mathbf{B}^{H}\mathbf{A}^{H}$ - **Unitary** $\mathbf{U}^{H}\mathbf{U}=\mathbf{I}$ - $U^{H}=U^{-1}$ - Eigenvectors form orthogonal basis of $\mathbb{C}^{n}$ - $|\det(\mathbf{U})|=1$ - Product of unitary matrices is unitary - Preserve angles between vectors and their lengths - **Orthogonal** $\mathbf{A}^{T}\mathbf{A}=\mathbf{A}\mathbf{A}^{T}=\mathbf{I}$ - $\mathbf{A}^{T}=\mathbf{A}^{-1}$ - Determinant is either $1$ or $-1$ - Real analog to unitary - **Upper Hessenberg** - Upper triangular matrix with additional element below diagonal