Numerical Matrix Analysis

Conjugates

Let $z=a+bi$ be a scalar with $a,b\in\Z$. Its complex conjugate $z^\ast$ is obtained by negating $b$ (if $b=0$ then $z$ is real and $z=z^\ast$): $$ z^\ast = a-bi. $$

Now let $A$ be a scalar $m\times n$ matrix: $$$ A = {[a_{ij}]}_{m\times n} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} $$$ Its hermitian conjugate (or adjoint) $A^\ast$ is the $n\times m$ matrix with ${ij}^\text{th}$ entry is the complex conjugate of the ${ji}^\text{th}$ entry of $A$ (i.e. of the form $a^\ast_{ij}=(a_{ji})^\ast={a_{ji}}^\ast$): $$$ A^\ast = {[a^\ast_{ij}]}_{n\times m} = \begin{pmatrix} a^\ast_{11} & a^\ast_{12} & \cdots & a^\ast_{1m} \\ a^\ast_{21} & a^\ast_{22} & \cdots & a^\ast_{2m} \\ \vdots & \vdots & \ddots & \vdots \\ a^\ast_{n1} & a^\ast_{n2} & \cdots & a^\ast_{nm} \end{pmatrix} = \begin{pmatrix} {a_{11}}^\ast & {a_{21}}^\ast & \cdots & {a_{m1}}^\ast \\ {a_{12}}^\ast & {a_{22}}^\ast & \cdots & {a_{m2}}^\ast \\ \vdots & \vdots & \ddots & \vdots \\ {a_{1n}}^\ast & {a_{2n}}^\ast & \cdots & {a_{mn}}^\ast \end{pmatrix} $$$ In the case that $A$ is an all-real matrix, then the adjoint of $A$ is simply the transpose of $A$ (i.e. $A^\ast=A^T$). Furthermore, if $A=A^\ast$ then $A$ is hermitian, and by definition must be a square matrix. (If $A$ all-real hermitian, $A=A^T$, and $A$ is said to be symmetric.)

Inner products

An inner product is a binary operation over a vector space which associates to a pair o