# Is there a matrix identity for $(I + \lambda B)^{-1}$, B full rank?

I need to invert the matrix $(I + \lambda B)^{-1}$ for several $\lambda$s. An identity of the form $(I + \lambda B)^{-1} = f(\lambda, B, B^{-1})$ would shortcut the costly matrix inversions. Any ideas?

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If the $\lambda$ are such that $\lVert\lambda B\rVert<1$ then you can use a series, namely $\sum_{k=0}^{+\infty}(-1)^k\lambda^k B^k$. –  Davide Giraudo Jun 24 '12 at 12:09
Would it be an option to use diagonalization of B? –  Gottfried Helms Jul 25 '12 at 7:04

I doubt there is such a shortcut, but perhaps some series can give you some practical approximation, using:

$$(I+A)^{-1}=I -A + A^2 - A^3 + \cdots$$

So

$$(I+\lambda B)^{-1}= I - \lambda B + \lambda^2 B^2 - \lambda^3 B^3 \cdots$$

which could be useful if $\lambda$ is small (as pointed out by Davide comment).

Otherwise, if $\lambda$ is large,you can instead use the identity $(I+A)^{-1}=A^{-1}(I+A^{-1})^{-1}$ and apply the same expansion:

$$(I+\lambda B)^{-1}= \beta C \, (I+\beta C)^{-1}= \beta C - \beta^2 C^2 + \beta^3 C^3 + \cdots \hspace{1cm} C=B^{-1} , \; \beta= \lambda^{-1}$$

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Food for thought thanks, but I don't know what the $\lambda$ values will be. $(1 + \lambda B)^{-1}$ is a term in an equation $g(\alpha, \lambda) = 0$ which is to be solved iteratively, each iteration requiring a new matrix inversion. I was looking for something along the lines of the Woodbury matrix identity, which fails here as B dense. Perhaps an if statement separating $\|\lambda B\| > 1$ and $\|\lambda B\| < 1$, with leonbloy's suggestion. –  newling Jun 24 '12 at 14:21

You can use the Sherman-Morrison-Woodbury formula to calculate the inverse.

The formula states that

$$(A+UCV)^{-1}=A^{-1}-A^{-1}C(C^{-1}+VA^{-1}U)^{-1}VA^{-1}$$

and you can obtain, recognising that $A=U=V=I$ and $C=\lambda B$,

$$(I+\lambda B)^{-1}=I-\lambda B\left(I+\frac{1}{\lambda}B^{-1}\right)^{-1}.$$

I'm not sure if that is helpful or not for your application if $B^{-1}$ is easier to operate with than $B$.

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Consider the characteristic polynomial $\chi_B$ of $B$ defined by $\chi_B(X)=\det(XI-B)$. The degree of $\chi_B$ is the size $n$ of $B$ and Cayley-Hamilton theorem says that $\chi_B(B)=0$.

For every scalar $x$, $\chi_B(X)=(X-x)Q_x(X)+\chi_B(x)$ for some polynomial $Q_x$ of degree $n-1$, hence $$\frac{\chi_B(x)}{X-x}=\frac{\chi_B(X)}{X-x}-Q_x(X).$$ In particular, for every $x$ such that $\chi_B(x)\ne0$, $(B-x)^{-1}=-\chi_B(x)^{-1}Q_x(B)$. Applying this to $x=-1/\lambda$ for some $\lambda\ne0$, one gets $$(I+\lambda B)^{-1}=\frac{Q_{-1/\lambda}(B)}{-\lambda\,\chi_B(-1/\lambda)},$$ The condition $\chi_B(-1/\lambda)\ne0$ is equivalent to the condition that $\lambda B+I$ is invertible hence this is not a restriction. Note that $Q_x$ is the polynomial $$Q_x(X)=\frac{\chi_B(X)-\chi_B(x)}{X-x}=\sum_{k=0}^{n-1}q^{(x)}_kX^k,$$ for some coefficients $q^{(x)}_k$, and that $$(I+\lambda B)^{-1}=\sum_{k=0}^{n-1}\frac{q^{(-1/\lambda)}_k}{q^{(-1/\lambda)}_0}B^k.$$ Note finally that one can simplify the computations by replacing $\chi_B$ by any polynomial of a lesser degree which is zero at $B$, for example the minimal polynomial $\mu_B$.

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