A real matrix is symmetric if . I will show in this post that a real symmetric matrix have real eigenvalues.
I will need a dot product for the prof and I’ll use the basic dot product for two vectors and : , where is the complex conjugate of the vector .
The useful property of this dot product is that , for any matrix .
And considering that is real, a simple proof is:
An eigenvalue have a correspondent eigenvector: .
We have and considering that A is symmetric .
From and because is not a zero vector it results that the imaginary part of is zero, so the eigenvalue is a real number.