Help Prove Real Eigenvalues of Symmetric Matrix

In summary, the conversation discusses the properties of symmetric matrices and how to prove that the eigenvalues of a row-normalized real symmetric matrix are also real numbers. The conversation also mentions an algorithm for determining if a matrix is symmetric or not.
  • #1
tom08
19
0
Help! Symmetric matrix

I know that all the eigenvalues of a real symmetric matrix are real numbers.
Now can anyone help out how to prove that "all the eigenvalues of a row-normalized real symmetric matrix are real numbers"? Thank you~~~
 
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  • #2


?? A "row normalized" symmetric matrix is still a symmetric matrix. Just use the same proof as for any symmetric matrix.
 
  • #3


Also note this thread.
 
  • #4


hi everyone please tell me an algorithm for finding if a matrix is symmetric or not
 
  • #5


What's wrong with the obvious one:

Answer= "symmetric"
For i= 1 to n-1
{
For j= i+1 to n
if [itex]a_{ij}\ne a_{ji}[/itex]
{
Answer= "not symmetric"
exit loops
}
}
report Answer.
 

What is a symmetric matrix?

A symmetric matrix is a square matrix that is equal to its own transpose. This means that the elements on either side of the main diagonal are equal to each other.

What are eigenvalues of a matrix?

Eigenvalues are the special values that represent the scaling factor of the eigenvectors when multiplied by a matrix. They are often used in linear algebra to solve systems of equations and to understand the behavior of a matrix.

Why is proving real eigenvalues of a symmetric matrix important?

Proving that the eigenvalues of a symmetric matrix are real is important because it allows us to make certain conclusions about the matrix. For example, a symmetric matrix with all real eigenvalues is always diagonalizable, which can simplify many calculations.

What are some methods for proving real eigenvalues of a symmetric matrix?

There are several methods for proving real eigenvalues of a symmetric matrix, including the Gershgorin circle theorem, Rayleigh quotient, and the spectral theorem. These methods use different properties of symmetric matrices to show that their eigenvalues are real.

Are there any special properties of symmetric matrices that make it easier to prove real eigenvalues?

Yes, there are a few special properties of symmetric matrices that make it easier to prove real eigenvalues. For example, symmetric matrices are always orthogonally diagonalizable, meaning they can be decomposed into a diagonal matrix and an orthogonal matrix. This can simplify calculations and make it easier to prove real eigenvalues.

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