Hermitian Matrix: Real & Imaginary Parts

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    Hermitian Matrix
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Discussion Overview

The discussion revolves around the properties of Hermitian matrices, specifically focusing on the implications of their real and imaginary parts. Participants explore the nature of diagonal elements in the context of Hermitian matrices and their relationships to symmetry and conjugation.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant notes that for a Hermitian matrix, the real part is symmetric and the imaginary part is anti-symmetric, questioning the implications for diagonal elements of the imaginary part.
  • Another participant asserts that the diagonal elements of a Hermitian matrix must be real numbers, implying that they cannot have non-zero imaginary parts.
  • A follow-up question seeks clarification on whether the diagonal elements are strictly zero or can be any real number, indicating a potential misunderstanding.
  • Further clarification is provided that the imaginary part of the diagonal elements must be zero for the elements to be their own conjugates, reinforcing that they are real numbers but not necessarily zero.

Areas of Agreement / Disagreement

Participants generally agree that the diagonal elements of a Hermitian matrix are real numbers, but there is some ambiguity regarding whether they can be zero or must be non-zero.

Contextual Notes

The discussion does not resolve the implications of the imaginary part being zero on the diagonal elements, leaving some assumptions about the nature of Hermitian matrices unexamined.

EngWiPy
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Hi,

Suppose that we have a complex matrix \mathbf{H} that is Hermitian. The real part of the matrix will be symmetric, and the imaginary part of the matrix will be anti-symmetric. But what about the diagonal elements in the imaginary part? I mean we deduce that the elements in the diagonal of the imaginary part of the matrix equal the negative of themselves! What does this mean? or am I wrong?

Thanks in advance
 
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y = - y implies y = 0 ... the diagonal elements of an hermitian matrix are real numbers.
 
Petr Mugver said:
y = - y implies y = 0 ... the diagonal elements of an hermitian matrix are real numbers.

Are you saying that the diagonal elements of a Hermitian matrix are zero, or real and could be zero?
 
S_David said:
Are you saying that the diagonal elements of a Hermitian matrix are zero, or real and could be zero?

He's saying that the imaginary part must be zero (for a number to be it's own conjugate). So yes he means real numbers not necessarily zero.
 
uart said:
He's saying that the imaginary part must be zero (for a number to be it's own conjugate). So yes he means real numbers not necessarily zero.

Ok, I got it. Thanks a lot.
 

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