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A Kac-Murdock-Szegö (KMS) matrix is a matrix of the form $A_{ij}=\rho^{|i-j|}$ for $i,j=1,2,\dots,n$ and $\rho\neq1$. The inverse ${\bf A}^{-1}$ is well known, see e.g., Dow's paper$^\color{magenta}{\dagger}$. However, I am interested in the inverse of a perturbed KMS matrix $\alpha {\bf I}_n + {\bf A}$, where ${\bf I}_n$ is the $n \times n$ identity matrix and $\alpha \in \mathbb{R} \setminus \{0\}$. Does anyone know an explicit formula (if there exists an explicit formula) for $\left(\alpha {\bf I}_n + {\bf A}\right)^{-1}$?

By "explicit formula", I am referring to a formula as for ${\bf A}^{-1}$, no formula like the Woodbury identity. At least a formula which involves only the computation of ${\bf A}^{-1}$ or $\bf A$ as

$$ (\alpha {\bf I}_n + {\bf A})^{-1} = \sum_{k=0}^{\infty} (-1)^k \alpha^{-k-1} {\bf A}^k $$

but with finite number of terms. Probably, the answer is negative.

$\color{magenta}{\dagger}$ Murray Dow, Explicit inverses of Toeplitz and associated matrices, 2002.

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Rodrigo de Azevedo
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user_goldeneye
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1 Answers1

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From Cayley-Hamilton, for $k \geq 0$, every power ${\bf A}^k$ can be written as a linear combination of the $n$ matrices ${\bf A}^0, {\bf A}^1, \dots, {\bf A}^{n-1}$. Thus,

$$ (\alpha {\bf I}_n + {\bf A})^{-1} = \sum_{k=0}^{n-1} c_k (\alpha) \, {\bf A}^k $$

and the only difficulty left is finding the coefficients $c_0 (\alpha), c_1 (\alpha), \dots, c_{n-1} (\alpha)$.

Rodrigo de Azevedo
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