# Proof of Blanchard Kahn conditions

I am trying to understand proof of BK conditions given in the appendix to their paper. In their paper authors have original model:

There $$X_{t + 1} \in \mathbb{R}^{n}, P_{t + 1} \in \mathbb{R}^{m}$$ This model is transformed into

by using the fact that $$A = C^{-1} \Lambda C$$. $$\Lambda$$ also is the matrix of eigenvalues of A, which are sorted in increasing order by modulus: $$|\lambda_{11}| < |\lambda_{22}| < ... < |\lambda_{(m+n)(m+n)}|$$. After constuction of $$\Lambda$$ we can decompose it into two submatrixes:

There $$\Lambda_1$$ consist of eigenvalues that are inside unit circle in $$\mathbb{C}$$. And $$\Lambda_2$$ consist of eigenvalues that are outside unit circle. By definition $$\Lambda_1$$ has $$\bar{n}$$ rows and columns. And $$\Lambda_2$$ has $$\bar{m}$$ rows and columns. It folows then that vector $$Y_t$$ may not have same amount of elements as $$X_t$$ because

and $$B_{11} \in \mathbb{R}^{n \times \bar{n}}$$. So $$Y_t$$ has $$\bar{n}$$ elements. We also know that $$n + m = \bar{n} + \bar{m}$$.

So there are my question: Appendix of paper Blanchard Kahn (1980) says that we can divide our original system of equations into two subsystems (yes, we can). Second subsystem will be $$E_t[Q_{t+1}] = \Lambda_2 Q_t + (C_{21} \gamma_1 + C_{22} \gamma_2) Z_t$$. It is also stated that system will explode unless:

$$Q_t = -\sum_{i=0}^{\infty} \Lambda_2^{-i-1}(C_{21}\gamma_1 + C{22} \gamma_2)E_t[Z_{t+i}] \quad(1)$$

However it does not seem to be obvious since we obtain (1) by just iterating formula $$E_t[Q_{t+1+i}] = \Lambda_2 Q_{t+i} + (C_{21} \gamma_1 + C_{22} \gamma_2) Z_{t + i}$$ for $$i=0,...,\infty$$. So how does non-explosiveness follows from (1)?

Expression (1) only follows from $$\mathbb{E}_t[Q_{t+1}]=\Lambda_2Q_t+(C_{21} \gamma_1 +C_{22}\gamma_2)Z_t$$ if we make an assumption about how fast $$Q_t$$ grows relative to $$\Lambda_2$$.
To see this, notice you can write the expression $$\mathbb{E}_t[Q_{t+1}]=\Lambda_2 Q_t +(C_{21}\gamma_1+C_{22}\gamma_2)Z_t$$ as $$Q_t=-\Lambda_2^{-1}(C_{21}\gamma_1+C_{22}\gamma_2)Z_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}]$$ This defines a recurrence relation of the form $$Q_t=\Lambda_2^{-1}B_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}]$$ where $$B_t=-(C_{21}\gamma_1+C_{22}\gamma_2)Z_t$$. Iterating this expression forwards \begin{align*} Q_t &= \Lambda_2^{-1}B_t+\Lambda_2^{-1}\mathbb{E}_t[Q_{t+1}] \\ &=\Lambda_2^{-1}B_t +\Lambda_2^{-1}\mathbb{E}_t[\Lambda_2^{-1}B_{t+1}+\Lambda_2^{-1}\mathbb{E}_{t+1}[Q_{t+1}]] \\ &= \Lambda_2^{-1}B_t +\Lambda_2^{-2}\mathbb{E}_t[B_{t+1}]+\Lambda_2^{-2}\underbrace{\mathbb{E}_t[\mathbb{E}_{t+1}[Q_{t+1}]]}_{=\mathbb{E}_t[Q_{t+1}]} \\ &=\Lambda_2^{-1}B_t +\Lambda_2^{-2}\mathbb{E}_t[B_{t+1}]+\Lambda_2^{-2}\mathbb{E}_t[Q_{t+1}] \\ & \ \ \vdots \\ &=\sum_{i=0}^{\infty}\Lambda_2^{-i-1}\mathbb{E}_t[B_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \\ &=\sum_{i=0}^{\infty}\Lambda_2^{-i-1}\mathbb{E}_t[-(C_{21}\gamma_1+C_{22}\gamma_2)Z_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \\ &=-\sum_{i=0}^{\infty}\Lambda_2^{-i-1}(C_{21}\gamma_1+C_{22}\gamma_2)\mathbb{E}_t[Z_{t+i}]+\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}] \end{align*}
Therefore, non-explosiveness corresponds to $$\lim_{\tau\to \infty}\Lambda_2^{-\tau-1}\mathbb{E}_t[Q_{t+\tau}]=0$$. As $$\Lambda_2$$ contains the eigenvalues outside the unit circle, $$\Lambda_2^{-\tau-1}$$ tends to the 0 matrix. Therefore, this condition is saying that for non-explosive behaviour we need $$\mathbb{E}_t[Q_{t+\tau}]$$ to grow at a rate slower than $$\Lambda_2^{\tau}$$. This limit condition is similar to the transversality condition which also precludes explosive paths.