Let $A$ be a symmetric matrix, of the following form: $$ A = \begin{bmatrix} a & b & 0 \\ b & c & 0 \\ 0 & 0 & d \end{bmatrix} $$ Furthermore, let $B$ be the transformation of this matrix, using the transformation matrix $P$: $$ B = P^TA P $$ The matrix $P$ is invertible but not necessarily orthogonal, i.e. $P^{-1}\neq P^T$. (Edit: However, the last column of P is a unit vector, i.e. it has length one).
I know the eigenvalues and eigenvectors of matrix $A$, but and I am interested in a relation with the eigenvalues and eigenvectors of matrix $B$. Since $P$ is not necessarily orthogonal, I cannot use matrix similarity, but I was thinking to use the Principle Axes Theorem, that states:
Principle Axes Theorem Let $A$ be an $n \times n$ symmetric matrix. Then there is an orthogonal change of variable, $\mathbf{x}=P\mathbf{y}$, that transforms the quadratic form $\mathbf{x}^TA\mathbf{x}$ into a quadratic form $\mathbf{y}^TD\mathbf{y} = \mathbf{y}^T P^T A P\mathbf{y}$ with no cross-product term. The columns of $P$ are the principle axes.
So to repeat my question: is there a relation between the eigenvalues and eigenvectors of $A$ and those of $B$?