Chapter 3: Q.7P (page 141)

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

Short Answer

Expert verified

$\begin{array}{ll} Identity Index Notation \\ A = \bar{A} A_{ij} = {\bar{A}}_{ij} \\ A = A^{⊤} A_{ij} = A_{ij}^{⊤} \\ A = - A^{⊤} A_{ij} = - A_{ij}^{⊤} \\ A^{- 1} = A^{⊤} {(A^{- 1})}_{ij} = A_{ij}^{⊤} \\ A = - \bar{A} A_{ij} = - {\bar{A}}_{ij} \\ A = A^{†} A_{ij} = A_{ij}^{†} \\ A = - A^{†} A_{ij} = - A_{ij}^{†} \\ A^{- 1} = A^{†} {(A^{- 1})}_{ij} = A_{ij}^{†} \\ {AA}^{†} = A^{†} A \sum_{k} A_{ik} A_{kj}^{†} = \sum_{k} A_{ik}^{†} A_{kj} \end{array}$

Step by step solution

Given information.

The given information is given below,

$A = \bar{A}$

$A = A^{⊤}$ , A is real symmetric.

$A = - A^{⊤}$ , A is real anti-symmetric.

$A^{- 1} = A^{⊤}$ , A is real orthogonal.

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

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

$A = - A^{†}$ , A is anti-Hermitian.

$A^{- 1} = A^{†}$ , A is unitary.

${AA}^{†} = A^{†} A$ , 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 $A = \bar{A}$ can be given as. $A_{i j} = {\bar{A}}_{i j}$

The index notation for $A = A^{⊤}$ can be given as $A_{ij} = A_{ij}^{⊤} .$

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

Short Answer

Step by step solution

Given information.

Index notation.

Write each item in index notation.

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