Finding the Inverse of Symmetric Matrices with Non-Real Coefficients

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SUMMARY

This discussion focuses on finding the inverse of symmetric matrices with non-real coefficients, specifically 3x3 matrices. The general formula for the inverse of a 3x3 matrix is referenced, with particular attention to the conditions for symmetric matrices where \(b=d\), \(c=g\), and \(f=h\). The example provided includes matrices with complex numbers, highlighting the need for a specialized approach to handle non-real coefficients.

PREREQUISITES
  • Understanding of matrix algebra, specifically 3x3 matrices
  • Familiarity with the concept of matrix inverses
  • Knowledge of symmetric matrices and their properties
  • Basic grasp of complex numbers and their operations
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  • Study the general formula for the inverse of a 3x3 matrix
  • Explore properties of symmetric matrices in linear algebra
  • Learn about matrix operations involving complex numbers
  • Investigate numerical methods for computing matrix inverses
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Mathematicians, students of linear algebra, and anyone working with complex matrices or seeking to understand the properties and inverses of symmetric matrices.

OhMyMarkov
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Hello everyone!

I'm struggling to find a general formula for obtaining an inverse of a symmetric matrix, for e.g.

1 i -1
i -i 2
-1 2 1

Any help is appreciated!
 
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OhMyMarkov said:
Hello everyone!

I'm struggling to find a general formula for obtaining an inverse of a symmetric matrix, for e.g.

1 i -1
i -i 2
-1 2 1

Any help is appreciated!

Hi OhMyMarkov, :)

Are you specifically concerned about 3x3 symmetric matrices? The general form of the inverse of a 3x3 matrix is given >>here<<. For a symmetric matrix \(b=d,\,c=g\mbox{ and } f=h\).

Kind Regards,
Sudharaka.
 
Hello Sudharaka!

Well, I'm actually interested in symmetric matrices that have this shape (the given example is for a 3x3 matrix):

1 1 1
1 2 4
1 4 16Note that the coefficients need not be real.
 

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