Relationship Between Hermitian and Symmetric Matrices

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Discussion Overview

The discussion revolves around the relationship between Hermitian and symmetric matrices, focusing on definitions and properties. Participants explore whether all symmetric matrices with real entries are Hermitian and vice versa, as well as the implications of complex entries in Hermitian matrices.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants question whether all symmetric matrices with real entries are Hermitian, suggesting a need to examine the properties of the conjugate transpose.
  • Others inquire about the relationship in the context of Hermitian matrices that may have complex entries, raising the question of whether the definition changes.
  • There is a definition provided for Hermitian matrices, stating that they are equal to their conjugate transpose, but the implications of this definition are still under discussion.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus on the relationship between symmetric and Hermitian matrices, with multiple viewpoints and questions remaining unresolved.

Contextual Notes

Some assumptions about the nature of matrix entries (real vs. complex) and the implications of the definitions are not fully explored, leaving room for further clarification.

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Are All symmetric matrices with real number entires Hermitian? What about the other way around-are all Hermitian matrices symmetric?
 
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Penemonie said:
Are All symmetric matrices with real number entires Hermitian?

What do you get when you take the conjugate transpose of a symmetric matrix with all real entries? Have you tried it?

Penemonie said:
What about the other way around-are all Hermitian matrices symmetric?

Are you this time including Hermitian matrices with complex (nonzero imaginary part) entries?
 
What is a Hermitian matrix, and what does this mean, if all entries were real?
 
fresh_42 said:
What is a Hermitian matrix, and what does this mean, if all entries were real?

A Hermitian matrix is a matrix that is equal to its conjugate transpose, i.e., ##H = H^{\dagger}##.
 

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