40 questions linked to/from Inverse of the sum of matrices
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### Effective way to calculate the inverse (A+kB)^-1 with k changing and A, B fixed

I have a Simulink modell where I need to calculate $(A+c_k B)^{-1}$ in every time step with $c_k$ changing each iteration. Does someone know any more effective way to do it, instead of calculating a ...
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### Matrix expression manipulation

Given the left hand side of the equation below, how to show the equality with the right hand side? I checked it numerically, but not sure how to prove it. \begin{align} A - AC^T(CAC^T +R)^{-1} CA = (...
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### Matrix function that gives a scalar

I have the following function: $$f(z) = z\vec{b}^T[I-zA]^{-1}\vec{1},$$ where $z$ is a complex scalar with $Re(z)<0$ (for simplicity, WLOG, we can take $z$ to be real), $b$ is a vector, $1$ is a ...
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### How to calculate $(I + GH)^{-1}$? [duplicate]

Suppose we have a matrix A. We have decomposed A into a sum of the identity matrix and the product of two column(G) and row(H) matrices. In general, how can we calculate $A^{-1}$?
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Let $S\in\mathbb{R}^{N\times N }$ be an orthogonal matrix and denote $S_{N,N-1}\in\mathbb{R}^{N \times N-1}$ as the matrix with the same elements of $S$ but without the last column of $S$. Let $A\in\... 0answers 27 views ### Reduced complexity of matrix inversion of sum of rank 1 matrices:$M^{-1} = \left( I + \sum \limits_{i=1}^{K} \alpha_i A_i \right)^{-1} $? How to reduce the complexity of matrix inversion of sum of rank 1 matrices, not only arithmetic but also run time?$M^{-1} = \left( I + \sum \limits_{i=1}^{K} \alpha_i A_i \right)^{-1} $where$A_i ...
How to show that $f(X)=tr(X^{-1})$ is convex over the set of symmetric real positive definite matrices by showing that $g(t)=f(X+tV)$ is convex for any such $X$ and $V$? Clearly we need to show that ...