I was deriving some equations for my projects and come to the following term, $$(\alpha Q+\beta I)^{-1}$$ where both $\alpha$ and $\beta$ are positive real numbers, $Q$ is an orthogonal matrix and $I$ is identity matrix. I have 2 questions:

  1. Is there any simplification for the inverse I can do knowing the property of the matrices?
  2. Can I approximate the sum as $(\alpha Q + \beta I) \approx \alpha Q$? How good is this approximation?

NB. For my application, $Q$ only needs to follow $Q^TQ=I$ i.e., only column orthogonality

  • $\begingroup$ Since $Q$ is square, $Q^T Q = I$ if and only if $QQ^T = I$. $\endgroup$ – eyeballfrog Aug 2 '19 at 15:03

Unfortunately your expression may not be invertible. For example, $$Q = \begin{bmatrix} 1 & 0 \\ 0 & -1\end{bmatrix} $$ and $\alpha = \beta$.

As for the approximation, notice that $||aQ + bI - aQ|| = ||bI|| = b$ using the operator norm. So if you would like your error to be smaller than $\epsilon$ then your $b$ better be such that $b < \epsilon$

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