Orthogonal and symmetric matrices

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SUMMARY

The discussion focuses on the formation of symmetric and orthogonal matrices, specifically addressing the conditions under which a matrix M can be both symmetric (M = MT) and unitary (MM† = I). The participants conclude that the identity matrix and diagonal matrices with entries of 1 or -1 are valid examples. Additionally, the distinction between symmetric and Hermitian matrices is emphasized, clarifying that the discussion pertains solely to real matrices.

PREREQUISITES
  • Understanding of matrix operations, including transpose and conjugate transpose.
  • Familiarity with the definitions of symmetric and orthogonal matrices.
  • Knowledge of unitary matrices and their properties.
  • Basic linear algebra concepts, particularly involving real matrices.
NEXT STEPS
  • Explore the properties of unitary matrices in detail.
  • Learn about the implications of symmetric matrices in linear transformations.
  • Investigate the relationship between eigenvalues and orthogonal matrices.
  • Study the construction of diagonal matrices and their applications in various mathematical contexts.
USEFUL FOR

Mathematicians, students of linear algebra, and anyone interested in advanced matrix theory and its applications in various fields such as physics and engineering.

mnb96
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Hello,
I guess this is a basic question.
Let´s say that If I am given a matrix X it is possible to form a symmetric matrix by computing X+X^{T}.

But how can I form a matrix which is both symmetric and orthogonal? That is:
M=M^{T}=M^{-1}.
 
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You have implicitly stated that M is real. In this case I think only the identity matrix matches your requirements.
 
And also diagonal matrix with 1 or -1 at diagonal. Any more?
 
Thanks for the answers.
I just noticed that unfortunately I stated my problem incorrectly.

Starting from a matrix, I wanted to find another matrix which is symmetric (not Hermitian!) and unitary. That is:

M=M^{T}
MM^\dagger=I

Here M^{T} means "transpose", while M^\dagger means "conjugate transpose".
 

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