# On summation of Orthogonal and diagonal matrix

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

• Since $Q$ is square, $Q^T Q = I$ if and only if $QQ^T = I$. – 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 $$b < \epsilon$$