Symbols:Linear Algebra

Symbols used in Linear Algebra

Determinant

$\begin {vmatrix} a & b \\ c & d \end {vmatrix}$

$\begin {vmatrix} a & b \\ c & d \end {vmatrix}$ denotes a determinant.

The $\LaTeX$ code for \(\begin {vmatrix} a & b \\ c & d \end {vmatrix}\) is \begin {vmatrix} a & b \\ c & d \end {vmatrix} .

Matrix

$\begin {pmatrix} a & b \\ c & d \end {pmatrix}$

$\begin {pmatrix} a & b \\ c & d \end {pmatrix}$ denotes a matrix.

The $\LaTeX$ code for \(\begin {pmatrix} a & b \\ c & d \end {pmatrix}\) is \begin {pmatrix} a & b \\ c & d \end {pmatrix} .

Inverse Matrix

$\mathbf A^{-1}$

$\mathbf A^{-1}$ denotes the inverse of the invertible matrix $\mathbf A$.

The $\LaTeX$ code for \(\mathbf A^{-1}\) is \mathbf A^{-1} .

Complex Conjugate

$\overline {\mathbf A}$

Symbol used for the complex conjugate of a matrix.

Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over the complex numbers $\C$.

Let $\overline {\mathbf A}$ denote the matrix formed from $\mathbf A$ by replacing each entry of $\mathbf A$ with its complex conjugate:

$\overline {\mathbf A} = \sqbrk \beta_{m n}: \forall i \in \set {1, 2, \ldots, m}, j \in \set {1, 2, \ldots, n}: \beta_{i j} = \overline {\alpha_{i j} }$

where $\overline {\alpha_{i j} }$ denotes the complex conjugate of $\alpha_{j i}$.

Then $\overline {\mathbf A}$ is the complex conjugate of $\mathbf A$.

The $\LaTeX$ code for \(\overline {\mathbf A}\) is \overline {\mathbf A} .

Hermitian Conjugate

$\mathbf A^\dagger$

Symbol used for the Hermitian conjugate of a matrix.

Let $\mathbf A = \sqbrk \alpha_{m n}$ be an $m \times n$ matrix over the complex numbers $\C$.

Then the Hermitian conjugate of $\mathbf A$ is defined and denoted:

$\mathbf A^\dagger = \sqbrk \beta_{n m}: \forall i \in \set {1, 2, \ldots, n}, j \in \set {1, 2, \ldots, m}: \beta_{i j} = \overline {\alpha_{j i} }$

where $\overline {\alpha_{j i} }$ denotes the complex conjugate of $\alpha_{j i}$.

That is, $\mathbf A^\dagger$ is the transpose of the complex conjugate of $\mathbf A$.

The $\LaTeX$ code for \(\mathbf A^\dagger\) is \mathbf A^\dagger .