Spectral norm of infinite sum of matrices

Question

The following arises in statistical time-series analysis but, as far as I can tell, my question is purely mathematical.

Suppose I have a positive definite matrix $C$ that satisfies $$ C = ACA' + V, $$

where $A$ has all eigenvalues less than one in absolute terms and $V$ is a positive definite matrix. I believe this is sufficient for $C$ to have the infinite sum representation $C = \sum_{k = 0}^{\infty}A^kV(A')^k$ based on what little I know about Lyapunov equations.

I want to assess the spectral norm of $C^{-1}$. For example, how does it relate to the spectral norm of $V$?

Answer 1

Since both $V$ and $C$ is symmetric and positive definite, we find for a vector $x$: $$ x^TCx =(Ax)^T C (Ax) + x^TVx \ge x^TVx. $$ Thus, the smallest eigenvalue $\lambda_n(C)$ of $C$ is $\ge$ the smallest eigenvalue $\lambda_n(V)$ of $V$. This implies $\|C^{-1}\|\le \lambda_n(C)^{-1}\le \lambda_n(V)^{-1}$: Let $y=C^{-1}x$, then $$ \|C^{-1}y\|^2 = \|x\|^2 \le \lambda_n(C)^{-1}x^TCx = \lambda_n(C)^{-1}y^TC^{-1}y, $$ hence $\|C^{-1}y\|\le \lambda_n(C)^{-1}\|y\|$. Thus, $$ \|C^{-1}\|\le \lambda_n(C)^{-1}\le \lambda_n(V)^{-1}. $$