Numerical Matrix Analysis

Conjugates

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

Now let AA be a scalar m×nm\times n matrix: A=[aij]m×n=(a11a12a1na21a22a2nam1am2amn) 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) AA^\ast is the n×mn\times m matrix with ijth{ij}^\text{th} entry is the complex conjugate of the jith{ji}^\text{th} entry of AA (i.e. of the form aij=(aji)=ajia^\ast_{ij}=(a_{ji})^\ast={a_{ji}}^\ast): A=[aij]n×m=(a11a12a1ma21a22a2man1an2anm)=(a11a21am1a12a22am2a1na2namn) 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 AA is an all-real matrix, then the adjoint of AA is simply the transpose of AA (i.e. A=ATA^\ast=A^T). Furthermore, if A=AA=A^\ast then AA is hermitian, and by definition must be a square matrix. (If AA all-real hermitian, A=ATA=A^T, and AA is said to be symmetric.)

Inner products

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