Does diagonalizable imply symmetric?

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In summary, not every diagonalizable matrix is symmetric, but a symmetric matrix is always diagonalizable. A non-symmetric matrix can be diagonalized if it has a full set of linearly independent eigenvectors. There is a relationship between diagonalizable and symmetric matrices, where a symmetric matrix is always diagonalizable but the converse is not true. To determine if a matrix is both diagonalizable and symmetric, you would need to check if it is equal to its own transpose and if it has a full set of linearly independent eigenvectors. Both diagonalizable and symmetric matrices have special properties, such as having real eigenvalues and orthogonal eigenvectors for symmetric matrices, and being able to represent linear transformations in a simpler form and having a unique
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In order to prove my PDE system is well-posed, I need to show that if a matrix is diagonalizable and has only real eigenvalues, then it's symmetric.



Homework Equations


I've found theorems that relate orthogonally diagonalizable and symmetric matrices, but is that sufficient?


The Attempt at a Solution


See above.
 
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  • #2
It's not true. Consider the matrix

[tex]\left(\begin{array}{cc} 1 & 2\\ 0 & 3\end{array}\right)[/tex]
 
  • #3
Darn.. well I guess to find another way to show the PDE is well-posed.
 

1. Does every diagonalizable matrix have to be symmetric?

No, not every diagonalizable matrix is symmetric. While a symmetric matrix is always diagonalizable, the converse is not necessarily true.

2. Can a non-symmetric matrix be diagonalized?

Yes, a non-symmetric matrix can be diagonalized as long as it has a full set of linearly independent eigenvectors.

3. What is the relationship between diagonalizable and symmetric matrices?

A symmetric matrix is always diagonalizable, but a diagonalizable matrix is not always symmetric.

4. How can I determine if a matrix is both diagonalizable and symmetric?

A matrix is symmetric if it is equal to its own transpose, and it is diagonalizable if it has a full set of linearly independent eigenvectors. So, to determine if a matrix is diagonalizable and symmetric, you would need to check if it is equal to its own transpose and if it has a full set of linearly independent eigenvectors.

5. Are there any special properties of diagonalizable and symmetric matrices?

Yes, symmetric matrices have many interesting properties, such as having real eigenvalues and orthogonal eigenvectors. Diagonalizable matrices also have some special properties, such as being able to represent linear transformations in a simpler form and having a unique matrix representation for each linear transformation.

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