In the book 'Orthogonal Polynomials of Several Variables' by Charles F. Dunkl and Yuan Xu page 11 is the picture below. I assume the matrix in the picture is related to
Corollary 1.3.10 For the case of orthonormal polynomials $p_n$, $xp_n(x)=a_{n-1}p_{n-1}(x)+b_n p_n(x)+a_n p_{n+1}(x)$ and $a_n=k_n/k_{n+1}=\sqrt{d_n d_{n+2}}/d_{n+1}$ where $k_n$ is coeff. leading or highest power of x and $b_n=\int_a^b xp_n(x)^2 d\mu(x)$
which is on the previous page. I understand the matrix identity but I can't understand why he says 'hence $P_n (x)$ = det$(xI_n-J_n)$ In the picture. I know it can be diagonalized by a similarity transform or by even a unitary matrix and have the same determinant. If one could easily show that the eigenvalues(which are the diagonal elements in the diagonalized matrix) were such that each had a maximum power of x for all different powers of x then I could understand that because eigenvectors of symmetric matrices are orthogonal. Note by using the upper case P in $P_n(x)$ it means that generally they are not normalized but instead I would assume they are such that the highest power of x has coefficient 1 or are monic polynomials ? Anyway can anyone explain because of that matrix identity one is supposed to understand. Like what is so special about that matrix identity have to do with orthogonality ? Note disregard that 'mouse' pointer symbol in first row of matrix.
from where does that $P_n(x)=...$ arise ?
Prior in his book on page 9 and 10 for the special case of monic orthogonal polynomials,his corollary 1.3.8 $P_{n+1}(x)=(x+B_n)P_n(x)-C_nP_{n-1}(x)$ where $C_n=\frac {d_{n+1}d_{n-1}}{d_n^2}$ which equals $a_{n-1}^2$ and $B_n=-\frac{d_n}{d_{n+1}}\int_a^b xP_n(x)^2d\mu(x)=-b_n=-\int_a^b xp_n(x)^2d\mu(x)$ where $p_n(x)$ is orthonormalized eg $\int_a^b p_n(x)^2d\mu(x)=1$, while monic $\int_a^b P_n(x)^2d\mu(x)=\frac{d_{n+1}}{d_n}$. So we have the same recurrence and assuming the initial starting values are the same it is unique. That is assume $P_0=1$ and note $d_0=1$ by his definition and $d_n=det(g_{ij})_{i,j=1}^n$ where $g_{i,j}$ are elements of the Gram matrix $<f_i,f_j>$