Why if i $P^t \cdot A \cdot P$ results in the dotproduct of the eigenvector times its eigenvalue?

Question

My exam question is:

The symmetric matrix $A$ has $5$ and $−2$ as the only eigenvalues. Furthermore the eigenspace corresponding to eigenvalue 5 is given:

$B$ Consider the quadratic form on $\mathbb{R}^3$ given by

$$Q(x) = xTAx$$

question: Calculate $Q(v)$

the solution they gave: $Q(v) = v^T A V = V^t 5 V = 5 v (dot) V = 15$

I don't understand why $A = 5$.

I played around with some and figured out if I fill in the eigenvector the result will be the dot product of the eigenvector times the corresponding eigenvalue.

can someone explain to me this?

Answer 1

$A\neq 5$. Like you said, $$Av=5v$$ That is the simplification they made.