Linked Questions

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47 views

Unique solution to a arbitrary non-linear system under monotonicity assumptions

I have a map $f:\mathbb{R}^n\times\mathbb{R}^m \to \mathbb{R}^n$ of two arguments $x, y$, which has a following properties: The jacobian matrix of $f$ wrt to the first argument $\frac{\partial f}{\...
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1answer
39 views

Prove that matrices $B$ and $(I+B)^{-1}$ commute

From following derivation, depending on step, where to substitute $A=I$ one can obtain: $$ (I+B)^{-1}=I-B(I+B)^{-1} \\ (I+B)^{-1}=I-(I+B)^{-1}B $$ From which follows that matrices $B$ and $(I+B)^{-...
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49 views

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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47 views

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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1answer
32 views

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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34 views

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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29 views

Inverse of $S_{N,N-1}'AS_{N,N-1}$ where $S$ is orthogonal and $A$ is of full rank.

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\...
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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 ...
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26 views

Convexity of a function over matrices [duplicate]

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 $$...
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Simplification of $x = \left(A + \sum_i \alpha_i B_i\right)^{-1} A \ y$, $A \succ 0, A \in M_{n,n}(\mathbb{C})$, $\textrm{rank}(B_i) = 1$

I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition. Problem definition: Say $A \...

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