Q12P Show that the definition of a He... [FREE SOLUTION]

Chapter 3: Q12P (page 141)

Show that the definition of a Hermitian matrix $(A = A^{t})$ can be writtenrole="math" localid="1658814044380" $a_{ij} = {\overset{炉}{a}}_{ji}$ (that is, the diagonal elements are real and the other elements have the property that $a_{12} = {\overset{炉}{a}}_{21}$ , etc.). Construct an example of a Hermitian matrix.

Short Answer

Expert verified

The Hermitian matrix is $A = A^{T}$

Step by step solution

Given information.

To show that the definition of a Hermitian matrix $(A = A^{*})$ can be written $a_{i j} = a_{j i} ($ that is, the diagonal elements are real and the other elements have the property that $a_{12} = a_{21}$ , etc.). Construct an example of a Hermitian matrix.

Hermitian matrix

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

Prove that the Hermitian matrix is A=AT.

By definition, $A = A^{T}$ .

Hence the complex conjugate transpose of any matrix A is equal to A.

A contains the element $a_{i j}$ .

Where,

represent rows and j represent columns.

Transpose means rows and columns getting interchanged.

Performing transpose operation,

localid="1658814788983" $(a_{i j}) = (a_{j i})$

Its complex conjugate is,

$(a_{i j}) = ({\overset{炉}{a}}_{j i})$ .

The example for Hermitian matrix is,

$A = (1 1 - i 2 1 + i 3 i 2 - i 0)$

Hence,

$A^{T} = A$

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Short Answer

Step by step solution

Given information.

Hermitian matrix

Prove that the Hermitian matrix is A=AT.

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91影视

Show that the definition of a Hermitian matrix (A=At)can be writtenrole="math" localid="1658814044380" aij=a炉ji(that is, the diagonal elements are real and the other elements have the property thata12=a炉21, etc.). Construct an example of a Hermitian matrix.

Short Answer

Step by step solution

Given information.

Hermitian matrix

Prove that the Hermitian matrix is A=AT.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Classical Mechanics

Electromagnetism

Oscillations

Solid State Physics

Electric Charge Field and Potential

Engineering Physics

Study anywhere. Anytime. Across all devices.