Is there a general form for the determinant of this matrix? This came up in trying to deal with small oscillations of an $N$-pendulum. I obviously want to calculate the characteristic polynomial in $\omega^2$ to see if I can deal with the equation even in special cases. Is it the first time you see this kind of matrix? Does the characteristic polynomial have a reasonably nice form? Oh, $N$ is the dimension of the matrix.
You can suppose $m,g,l>0.$
$$A=\begin{pmatrix}
 N(mgl-\omega^2ml^2) & -\omega^2ml^2(N-1)&  -\omega^2ml^2(N-2)&...&   -\omega^2ml^2 \\ 
 -\omega^2ml^2(N-1)& (N-1)(mgl-\omega^2ml^2)  &-\omega^2ml^2(N-2)  &...  &-\omega^2ml^2 \\ 
 -\omega^2ml^2(N-2)   &-\omega^2ml^2(N-2)  &(N-2)(mgl-\omega^2ml^2)  & ...& -\omega^2ml^2 \\ 
\vdots&\vdots&\vdots&\vdots&\vdots
\end{pmatrix}$$
Thank you in advance.
EDIT: As pointed out by Omnomnomnom (I had to copy paste that name) the problem reduces to finding the determinant of:
$$A = -\omega^2ml^2 
\pmatrix{
N&N-1&N-2&\cdots&1\\
N-1&N-1&N-2&\cdots&1\\
N-2&N-2&N-2&\cdots&1\\
\vdots&\vdots&\ddots
}
+ mgl
\pmatrix{
N\\&N-1\\&&N-2\\&&&\ddots
}=-\omega^2ml^2T+mglV$$
Dividing by $mgl$ and calling $$\tilde\omega=\frac{\omega^2l}{g}$$
We have
$$\operatorname{Det}\bigg(\frac{A}{mgl}\bigg)=\operatorname{Det}B=\operatorname{Det}(V-\tilde\omega T)$$
I computed the first few polynomials in $\tilde \omega$.
For $N=2$
$$P_2(\tilde\omega)= \tilde\omega^2-4\tilde\omega+2$$
For $N=3$:
$$P_3(\tilde\omega)=-\tilde\omega^3+9\tilde\omega^2-18\tilde\omega+6$$
For $N=4$
$$P_4(\tilde\omega)=\tilde\omega^4-16\tilde\omega^3+72\tilde\omega^2-96\tilde\omega+24$$
For $N=5$
$$P_5(\tilde\omega)= -\tilde\omega^5+25\tilde\omega^4-200\tilde\omega^3+600\tilde\omega^2-600\tilde\omega+120$$
Do you see a pattern? The first coefficient might be $(-1)^N$, the second $N^2(-1)^{N+1}$, the last one is probably $N!$, the second to last one might be $-N!N$.

Consider the matrix
$$
P^{-1}Q = 
\pmatrix{
N\\&N-1\\&&N-2\\&&&\ddots
}^{-1}
\pmatrix{
N&N-1&N-2&\cdots&1\\
N-1&N-1&N-2&\cdots&1\\
N-2&N-2&N-2&\cdots&1\\
\vdots&\vdots&\ddots
} = \\
\pmatrix{
1&1-1/N&1-2/N&\cdots&1/N\\
1&1&1 - 1/(N-1)&\cdots&1/(N-1)\\
1&1&1&\cdots&1/(N-2)\\
\vdots&\vdots&\ddots
}
$$
Let $q_N(x)$ denote the characteristic polynomial $\det(P^{-1}Q - xI)$.
The answer to the determinant we seek will be
$$
\det(A) = N! \cdot a^N q_N(-b) = 
N! (-\omega^2 m \ell^2)^N q_N(- mg \ell)
$$
Hopefully you find this expression useful.
