1. ## 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

2. Hello,
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.

3. 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!

4. 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