# Spectral decomposition Intution

When $A$ is symmetric we can write it as $A = UDU^T$ where $D$ is diagonal. What I know is $A$ can be seen as diagonal matrix in some basis .

Is this "some basis" defined by orthogonal eigenvectors?

You forgot to mention that $U$ can be taken to be an orthogonal matrix. Then $U^T=U^{-1}$, and $A$ is obtained from $D$ by a change of basis according to $U$. So your "orthogonal basis" is formed by the columns of$~U$. When expressed on this basis the linear operator becomes diagonal, therefore these basis vectors are indeed eigenvectors.