existence of polynomial equation system solution

2012-01-26T07:16:23Z

For $1 \leq i \leq n$, let $A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \\ \end{bmatrix}$

$B_i=\begin{bmatrix} b_{i1} \\ \vdots \\ b_{in} \end{bmatrix}$ and $C_i=\begin{bmatrix} c_{i1} \\ \vdots \\ c_{in} \end{bmatrix}^*$

Let $D=A+\sum_{1 \leq i \leq n}B_i k_i C_i$. Then, for almost all $a_{ij},b_{ij},c_{ij}$, there exists $k_i \in \mathbb{C}$ such that all eigenvalues of $D$ are zeros.

Answer by Gottfried Helms for existence of polynomial equation system solution

2012-01-26T21:20:16Z

[update] By an example of 4x4-matrices the ansatz below could not be used to solve the problem. The matrix $\small Q_K $ cannot in general be made lower triangular by choices of the $\small k_i $. I'll delete this answer soon if I cannot improve the ansatz.

I do not yet have a full answer but possibly a first step into one. I think that this can be answered unsing two facts:

a) There is a similarity transformation with a rotation T such that $\small P = T^{-1}\cdot A \cdot T = T' \cdot A \cdot T $ where P is triangular and has the eigenvalues of A on its diagonal.

b) Your sum-expression of vector outer products, let it be called matrix $\small E_K = \sum_{i=1}^n B_i k_i C_i = \sum_{i=1}^n k_i (B_i C_i)= \sum_{i=1}^n k_i E_i $ is a weighted (by the $\small k_i $ weights) sum of rank-1-matrices $\small E_i $

From the "similarity rotated version" of all matrices

$\qquad \small Q_K = T' E_K T = \sum_{i=1}^n k_i (T' E_i T) = \sum_{i=1}^n k_i Q_i $ (which should be made triangular by choices of $\small k_i $ ) and
$\qquad \small R = T' D T $ which is then also triangular

we get your final equation in its form with triangular matrices

$\qquad \small R = P + Q_K $

We'll have a solution if the weights $\small k_i $ for the non-triangular, generic but rank-1-matrices $\small Q_i $ can be chosen such that their sum $\small Q_K$ becomes triangular and its diagonal equals the negative diagonal in $\small P $.

I've a vague speculation that the equation with this triangular matrices can easier be shown to be "almost always" possible, but have not yet a further concrete approach. At least this construction exhibits that the rank-1-matrices $\small E_i $ (and thus $\small Q_i$ ) cannot be simple scalar multiples of each other if A has full rank and thus the restriction to "for almost all" is unavoidable and possibly mainly consists of this property.

existence of polynomial equation system solution - MathOverflow

existence of polynomial equation system solution

Answer by Gottfried Helms for existence of polynomial equation system solution