Diagonalization of Specif Matrix

[ApproxIdent]

Now this could seem like a homework problem...but it's not. (I guess you'd need to believe me or just choose not to answer my question.) I'm trying to compute the eigenvalues of a matrix and it's a little more irritating than I'd expected. All I really care is if they're positive (so all I really need is for the top determinant to be positive). The matrix is

$\begin{bmatrix} 1 & 0 & 0 & \cdots & v_1 \\ 0 & 1 & 0 & \cdots & v_2 \\ 0 & 0 & 1 & \cdots & v_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ v_1 & v_2 & v_3 & \cdots & v_n \end{bmatrix}$

I feel like there must be some basic way of doing this that I'm missing. I'm reading a paper that works with (basically) the same matrix. The author writes down specific conditions (hich I presume is correct) so I could just accept it on face, but I'm a little frustrated at my inability to do this myself.

Anyway, if you're the type to love solving these problems you could spare me a bit of a headache...thanks for any help.

 

[ApproxIdent]

Never mind...it was easy. Apparently the process of deleting rows and columns is too complex for me. In case you're curious, it's:

$v_n - \sum_{i=1}^{n-1} v_i^2$