How to make the eigenvalue of this matrix inside or on the unit circle?

Question

Vector autoregressive process is defined by: $$X_{t} = \sum_{p=0}^{P}A_pX_{t-p}+\epsilon_t$$ I want to know how to generate the real value $A_p$s so that the eigenvalue of the matrix \begin{bmatrix} A_1 & A_2 & \dots & A_p \\ I & O & \dots & O \\ O & I & \dots& O \\ \vdots & \vdots & \vdots & \vdots \end{bmatrix} are all inside or on the unit circle. I tried to divide the $A_p$s by the maximum eigenvalue of $\sum A_p$ so that 1 is an eigenvalue of this matrix but there is no guarantee that 1 is the largest eigenvalue. Thanks in advance.

Answer 1

If you want to randomly generate such matrices, then you can proceed as follows. We construct the $(a_i)$ s.t. $\rho(A)=\alpha$ where $\alpha >0$ is given.

Note that the characteristic polynomial of $A$ is $p(x)=x^n-a_1x^{n-1}-a_2x^{n-2}-\cdots$.

1. You randomly choose the list $V=(a_i)$. For example (if $n=4$), let

$V=[4, 2, -5, -9]$.

2. You calculate the associated $\rho(A)$; here $\rho(A)\approx 4.054$.

3. Let $q(x)=(\dfrac{\alpha}{\rho(A)})^np(\dfrac{\rho(A)}{\alpha} x)$. For $\alpha =\rho(A)/4$, we obtain $q(x)=x^4-x^3-(1/8)x^2+(5/64)x+9/256$.

4. Finally we obtain for the $(a_i)$: $[1,1/8,-5/64,-9/256]$.

Note that there are well-known polynomials that have all their roots in the unit disc, but they are not random.

EDIT. Assume that the $(A_j)$ are $k\times k $ matrices. We want to obtain a matrix s.t. $\rho(A)\approx 1$.

1. Randomly choose the $(A_i)$.

   2. Calculate $\rho(A)$.

   3. Iterate that follows

$A_1=\dfrac{1}{\rho(A)}A_1,A2=\dfrac{1}{\rho(A)^{2}}A_2,A_3=\dfrac{1}{\rho(A)^{3}}A_3,\cdots$.

Calculate $\rho(A)$.