# Symmetric Matrix

• September 22nd 2008, 10:20 PM
Chris0724
Symmetric Matrix
The theorem state that the eigenvalues of a symmetric are real.

[ 3 4 | has real eigenvalues 1 & 5 (verified) but why not symmetric?
| 1 3 |

Rgds
• September 22nd 2008, 10:57 PM
Moo
Hello,
Quote:

Originally Posted by Chris0724
The theorem state that the eigenvalues of a symmetric are real.

[ 3 4 | has real eigenvalues 1 & 5 (verified) but why not symmetric?
| 1 3 |

Rgds

This is logic ! :)

$\text{Symmetric matrix } \implies \text{ Real eigenvalues}$

But this is different from $\text{Real eigenvalues } \implies \text{ Symmetric matrix}$.

This is the converse of the implication. And is not always true.
• September 22nd 2008, 11:08 PM
Chris0724
Quote:

Originally Posted by Moo
Hello,

This is logic ! :)

$\text{Symmetric matrix } \implies \text{ Real eigenvalues}$

But this is different from $\text{Real eigenvalues } \implies \text{ Symmetric matrix}$.

This is the converse of the implication. And is not always true.

So in short, i will always get a Real eigenvalues from a symmetric matrix but Real eigenvalues does not meant that the matrix must be symmetric?

Thanks!
• September 22nd 2008, 11:11 PM
Moo
Quote:

Originally Posted by Chris0724
So in short, i will always get a Real eigenvalues from a symmetric matrix but Real eigenvalues does not meant that the matrix must be symmetric?

Thanks!

Exactly !

The symmetry of a matrix is a sufficient but not necessary condition. If it was, you would have an equivalence, not an implication :)

Necessary and sufficient condition - Wikipedia, the free encyclopedia