eigenvectors of A are mutually orthogonal as conjugate with A?

Question

i read below statment in "Numerical Optimization by Jorge Nocedal" But I could not understand Can anyone guide why this is always true?

statement:

the eigenvectors $v_1,...,v_n $ of A are mutually orthogonal as well as conjugate with respect to A?

nonzero vectors $v_1,...,v_n \ \ $is said to be conjugate with respect to the symmetric positive definite matrix A if $\ \ v_j^TAv_j=0\ \ \forall\ \ i\neq j$

Answer 1

Let $v_i$ have eigenvalue $\lambda_i$. If $v_i$ and $v_j$ are orthogonal, then $v_i^tv_j=0$. But also $v_i^tAv_j=v_i^t(\lambda_j v_j)=\lambda_j v_i^tv_j=0$.