What can we say about $(I-AD)^{-1}$ if $D$ is a diagonal matrix?

Assume we know that square matrices $A$ and $(I-AD)^{-1}$ are invertible and also $D$ is a diagonal matrix. Also assume that $A$ is a symmetric matrix. My question is when we can express $(I-AD)^{-1}$ as a function of $A, D, A^{-1}, D^{-1}, (I-A)^{-1}$ (without terms $(A-D)^{-1}$ or $(A^{-1}-D)^{-1}$) ?

For example when matrix A is rank 1, then we have:

$(I-AD)^{-1}=I+\frac{1}{1-tr(AD)} AD$. As you can see if A is rank 1 then we can do this easily.

The only related paper I found is a paper by Kenneth S. Miller, but it is not useful for higher rank matrices.

I know it might be very hard for general matrix $A$ but can it be done for special cases where for instance matrix A is positive semidefinite? Any comment is highly appreciated.

• – Inquest Feb 13 '13 at 23:34
• Thanks for pointing this out, but I believe by using this theorem still term $(I-AD)^{-1}$ will show up. – user54626 Feb 14 '13 at 0:23
• It's not clear to me what you mean by "a function of $A,D,A^{-1},...$ etc, because $(I - AD)^{-1}$ to me is already written as a function of those things. – Christopher A. Wong Feb 14 '13 at 0:36
• For example, we can rewrite this as $(A^{-1} - D)^{-1}A^{-1}$, but that doesn't exactly simplify anything in particular. – Christopher A. Wong Feb 14 '13 at 0:37
• I'm not sure what your context is, since you are mentioning $(I-A)^{-1}$ while the invertibility of this matrix does not follow from your assumption. Would you consider something along the line of $(I - AD)^{-1} = I + (AD) + (AD)^2 + \cdots$ whenever $||AD|| < 1$ as a satisfactory answer? – user27126 Feb 16 '13 at 5:40

I have an answer to the following question : Is there a four-variable (not necessarily commutative) polynomial $f$ such that the identity

$$(I-AD)^{-1}=f((I-A)^{-1},A,D,A^{-1},D^{-1}) \tag{1}$$

holds, whenever $A$ is symmetric positive definite, and $D$ is invertible and diagonal ?

The answer is NO. Indeed, this is already impossible when $n=2$ and $$D=\left(\begin{matrix} 2 & 0 \\ 0 & 3 \end{matrix}\right).$$ If we write

$$A=\left(\begin{matrix} a & b \\ b & a \end{matrix}\right)$$

with $a\gt 0$ and $a\gt b$, then

$$\det(A)=a^2-b^2, \det(I-A)=a^2-2a-b^2+1, \det(I-AD)=6a^2-5a-6b^2+1$$

Thus the RHS in (1) will always have a denominator of the form

$$(a^2-b^2)^p (a^2-2a-b^2+1)^q,$$

where $p$ and $q$ are integers. Now the LHS in (1) will always a denominator of the form $(6a^2-5a-6b^2+1)^r$. Since we have three distinct irreducible polynomials in ${\mathbb Q}[a,b]$ here, the denominators will never coincide.

• Very nice, I'm curious if it's possible to see the infinite series solution 1/(1-x) = 1 + x + ... Somehow in determinant polynomial space? – kbb Feb 19 '13 at 22:29

In his book Some Eclectic Matrix Theory, Kenneth S. Miller presents the following generalization of your rank-$$1$$ result. This result is handy if your diagonal matrix $$D$$ is low rank.

Theorem (Miller, p. 14): Let $$H$$ be an $$n\times n$$ real matrix or rank $$r$$, and let $$\sigma_1, \ldots, \sigma_n$$ be the elementary symmetric functions of its eigenvalues. If $$I+H$$ is nonsingular then $$a_r \ne 0$$ and $$(I+H)^{-1} = I-\frac{1}{a_r}\big(a_{r-1}H - a_{r-2}H^2 + \cdots + (-1)^{r-1}a_0 H^r\big)$$ where $$a_m = 1 + \sum_{1 \le k \le m} \sigma_k, \quad m = 0,1,\ldots,r.$$

As for the elementary symmetric functions of the eigenvalues: They can be computed (via the Newton–Girard identities) from the power sums of the eigenvalues, which themselves are simply the traces of the powers of $$H$$ (as can be seen from writing $$H$$ in Jordan form): $$\operatorname{tr}(H^m) = \sum_{i=1}^n \lambda_i^m, \quad m \ge 0.$$ The upshot is that $$(I+H)^{-1}$$ is a degree-$$\operatorname{rk}(H)$$ polynomial in $$H$$ whose coefficients are rational functions in the traces of the powers of $$H$$ (Miller, Some Eclectic Matrix Theory, pp. 14-15).