Chapter 3: Q23P (page 142)

Show that the following matrices are Hermitian whether Ais Hermitian or not: $A A^{t}, A + A^{t}, i (A - A^{t})$ .

Short Answer

Expert verified

The $i (A - A^{T})$ is the Hermitian matrix.

Step by step solution

Given information.

The given matrices are:

_{$A A^{T}, A + A^{T}, i (A - A^{T})$}

Hermitian matrix and condition for a matrix to be Hermitian.

A Hermitian matrix is one in which the conjugate transpose of a complex square matrix is equal to itself.

Any square matrix Ais said to be Hermitian. If,
$A^{*} = A^{T}$

Where Ais the complex conjugate and $A^{T}$ is the transpose of A.

Show that the first and second equations are Hermitian or not.

Solve for the first case.

$({(A A^{T})}^{T} = {(A^{T})}^{T} A^{T})$

Since,

${(A B)}^{T} = B^{T} A^{T} = A A^{T} (A A^{T}) = A A^{T}$

Hence, $A A^{T}$ is the Hermitian matrix

Now solve for the second case:

$(A + A^{T}) = A^{T} + {(A^{T})}^{T}$

Since,

${(A + B)}^{T} = A^{T} + B^{T}$

Then,

$A^{T} + A = A^{T}$

Hence,

$(A + A^{T}) = A + A^{T}$

Therefore, the $A + A^{T}$ is the Hermitian matrix.

Show that the third equation is Hermitian or not.

Now solve the third equation:

${(i {(A - A)}^{T})}^{T} = i A^{T} - {(A^{T})}^{T}$

Since,

$(∵ {(A - B)}^{T} = A^{T} - B^{T})$

Then,

$i (A^{T} - {(A^{T})}^{T}) = - i (A^{T} - A) (∵ \bar{i} = - i) \bar{i} A^{T} - (A^{T}) = i (A - A^{T}) {(i {(A - A)}^{T})}^{T} = i (A - A^{T})$

Therefore, the $i (A - A^{T})$ is the Hermitian matrix.

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Show that the following matrices are Hermitian whether Ais Hermitian or not: $A A^{t}, A + A^{t}, i (A - A^{t})$ .

Short Answer

Step by step solution

Given information.

Hermitian matrix and condition for a matrix to be Hermitian.

Show that the first and second equations are Hermitian or not.

Show that the third equation is Hermitian or not.

One App. One Place for Learning.

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Show that the following matrices are Hermitian whether Ais Hermitian or not:AAt,A+At,i(A-At) .

Short Answer

Step by step solution

Given information.

Hermitian matrix and condition for a matrix to be Hermitian.

Show that the first and second equations are Hermitian or not.

Show that the third equation is Hermitian or not.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Quantum Physics

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Study anywhere. Anytime. Across all devices.

Show that the following matrices are Hermitian whether Ais Hermitian or not: $A A^{t}, A + A^{t}, i (A - A^{t})$ .