Relationship Between Hermitian and Symmetric Matrices

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SUMMARY

All symmetric matrices with real number entries are indeed Hermitian matrices, as they satisfy the condition of being equal to their conjugate transpose. However, not all Hermitian matrices are symmetric, particularly when they contain complex entries. A Hermitian matrix is defined as a matrix that equals its conjugate transpose, denoted as ##H = H^{\dagger}##. This distinction is crucial for understanding the properties and applications of these matrix types in linear algebra.

PREREQUISITES
  • Understanding of matrix operations, specifically conjugate transposition
  • Familiarity with the definitions of symmetric and Hermitian matrices
  • Knowledge of complex numbers and their properties
  • Basic linear algebra concepts, including eigenvalues and eigenvectors
NEXT STEPS
  • Research the properties of Hermitian matrices in complex vector spaces
  • Explore applications of symmetric and Hermitian matrices in quantum mechanics
  • Learn about eigenvalue decomposition for Hermitian matrices
  • Study the implications of matrix symmetry in optimization problems
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Students and professionals in mathematics, physics, and engineering, particularly those focusing on linear algebra, quantum mechanics, and matrix theory.

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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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