# Eigenvalue decomposition of non symmetric matrix

Often in examples, eigenvalue decomposition $A=U\Lambda U^T$, $A$ is usually assumed to be a symmetric matrix. I am wondering what are the differences and implications when $A$ is a non-symmetric (still positive values if that helps).

What can we say about the eigenvalues and eigenvectors of such decomposition?

The first implication of symmetry is normality. All Matrix, that suffice $A^TA = AA^T$ are acalled normal and the eigenvectors are orthogonal to each other. This makes it possible, to write $A = U Λ U^T$ instead of $A = U Λ U^{-1}$, which is correct for diagonizable matrices.
This means, that either $Λ \in \mathbb{C}^{n \times n}$ or you get $2\times 2$ blocks, instead of a diagonal matrix.