Question

Write each of the items in the second column of (9.2)in index notation.

Short answer

$\begin{array}{l}\text{Identity}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{Index Notation}\\ \mathrm{A}=\overline{\mathrm{A}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{A}}_{\mathrm{ij}}={\overline{\mathrm{A}}}_{\mathrm{ij}}\\ \mathrm{A}={\mathrm{A}}^{\mathrm{\top }}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{A}}_{\mathrm{ij}}={\mathrm{A}}_{\mathrm{ij}}^{\mathrm{\top }}\\ \mathrm{A}=-{\mathrm{A}}^{\mathrm{\top }}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{A}}_{\mathrm{ij}}=-{\mathrm{A}}_{\mathrm{ij}}^{\mathrm{\top }}\\ {\mathrm{A}}^{-1}={\mathrm{A}}^{\mathrm{\top }}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\left({\mathrm{A}}^{-1}\right)}_{\mathrm{ij}}={\mathrm{A}}_{\mathrm{ij}}^{\mathrm{\top }}\\ \mathrm{A}=-\overline{\mathrm{A}}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{A}}_{\mathrm{ij}}=-{\overline{\mathrm{A}}}_{\mathrm{ij}}\\ \mathrm{A}={\mathrm{A}}^{†}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{A}}_{\mathrm{ij}}={\mathrm{A}}_{\mathrm{ij}}^{†}\\ \mathrm{A}=-{\mathrm{A}}^{†}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\mathrm{A}}_{\mathrm{ij}}=-{\mathrm{A}}_{\mathrm{ij}}^{†}\\ {\mathrm{A}}^{-1}={\mathrm{A}}^{†}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\left({\mathrm{A}}^{-1}\right)}_{\mathrm{ij}}={\mathrm{A}}_{\mathrm{ij}}^{†}\\ {\mathrm{AA}}^{†}={\mathrm{A}}^{†}\mathrm{A}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\sum _{\mathrm{k}} {\mathrm{A}}_{\mathrm{ik}}{\mathrm{A}}_{\mathrm{kj}}^{†}=\sum _{\mathrm{k}} {\mathrm{A}}_{\mathrm{ik}}^{†}{\mathrm{A}}_{\mathrm{kj}}\end{array}$

Step-by-step solution

Given information.

The given information is given below,

$\mathrm{A}=\overline{\mathrm{A}}$

$\mathrm{A}={\mathrm{A}}^{\mathrm{\top }}$, A is real symmetric.

$\mathrm{A}=-{\mathrm{A}}^{\mathrm{\top }}$, A is real anti-symmetric.

$A^{-1}=A^{\mathrm{\top }}$, A is real orthogonal.

$\mathrm{A}=-\overline{\mathrm{A}}$, A is pure imaginary.

$A=A^{+}$, A is Hermitian.

$\mathrm{A}=-{\mathrm{A}}^{†}$, A is anti-Hermitian.

$A^{-1}=A^{†}$, A is unitary.

${\mathrm{AA}}^{†}={\mathrm{A}}^{†}\mathrm{A}$, normal.

Index notation. 

A matrix is written with two indices, say i and j, separated by commas or not: aij or ai,j , where the first subscript is the row number and the second subscript is the column number.

Write each item in index notation.

The index notation for$A=\overline{A}$ can be given as.$A_{ij}={\overline{A}}_{ij}$

The index notation for$\mathrm{A}={\mathrm{A}}^{\mathrm{\top }}$can be given as${\mathrm{A}}_{\mathrm{ij}}={\mathrm{A}}_{\mathrm{ij}}^{\mathrm{\top }}.$