39 questions linked to/from Inverse of the sum of matrices
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### How to inverse $(I + \alpha M)$ for all $\alpha$ [duplicate]

I am looking for a way to solve the following equation: $$(I + \alpha M)X=F$$ $\forall \alpha \in R$ (the real domain), with $M$ a square complex matrix without any particular properties, $I$ the ...
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### What is inverse of $I+A$?

Assume $A$ is a square invertible matrix and we have $A^{-1}$. If we know that $I+A$ is also invertible, do we have a close form for $(I+A)^{-1}$ in terms of $A^{-1}$ and $A$? Does it make it any ...
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### Inverse of symmetric matrix plus identity matrix

Consider the symmetric, positive definite matrix $\mathbf{A}$. I'd like to find a general form for $$(\mathbf{I} + \mathbf{A})^{-1}$$ that only involves $\mathbf{A}^{-1}$, i.e., no other inverse ...
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So before we begin, I already know the answer. I'm just having difficulty figuring out the steps for finding it. Given $u,v \in \mathbb{R}^{n}$, I want to show that $$(I+uv^{T})^{-1}= I - \frac{uv^{T}... 3answers 1k views ### Least Squares with Euclidean ( {L}_{2} ) Norm Constraint Suppose I have set of samples (x_i,y_i), 1 \leq i \leq n. I am interested in solving the following optimization problem:$$ \min \sum_{i=1}^n (y_i-a^\top x_i)^2, \quad \text{s.t } \|a\|_{2} = 1. $$... 1answer 654 views ### Is there a way to solve explicitly the following functional equation? I want to find an unknown function (actually CDF) F(p) which solves 1 - \lambda F(\frac{q_L}{q_H}p) - (1-\lambda)F(p-[q_H-q_L]) - \frac{K}{p-c_H} = 0, where 0<\lambda<1, q_H > q_L &... 2answers 375 views ### Proof of matrix inverse Prove that (A+uv^T)^{-1}=A^{-1}-\frac{1}{1+v^TA^{-1}u}A^{-1}uv^TA^{-1} Can someone give a hint how to show it.I am not getting from where to start. 1answer 938 views ### Inverse of sum of two marices, one being diagonal and other unitary. C = A+D, A being square matrix and D a full rank diagonal matrix. Is there any easy way to compute C^{-1} from A^{-1} and D Edit 2: (important edit) Iam interested in this question, ... 2answers 590 views ### Inverse of a sum of PSD matrices I was wondering if anyone knew any techniques to convert the following:  (A+B+C+..)^{-1}  where A,B,C... are positive semi-definite (PSD) matrices into a sum of some other function:  f(A)+f(B)+... 2answers 536 views ### Approximating the inverse of a perturbed matrix Consider a matrix A which we subject to a small perturbation \partial A. If \partial A is small, then we have (A + \partial A)^{-1} \approx A^{-1} - A^{-1} \partial A A^{-1} I came across ... 1answer 489 views ### How to convert (A+\lambda E)^{-1}? Here is one of the most famous equation called Sherman–Morrison formula (1951) when we want to get an inverse matrix.$$(A+vw^{\text{T}})^{-1}=A^{-1}-\cfrac{A^{-1}vw^{\text{T}}A^{-1}}{1+{w}^{\text{T}}...
I need a way to compute the inverse of the sum of three matrices: $(A + BB^T + \beta I)^{-1}$ where $I$ is identity and $\beta$ is a constant. I am not very familiar with linear algebra, but a ...
### Find two $2\times2$ real matrix $A$ and $B$ such that $A$, $B$ , $A+B$ are all invertible with $(A+B)^{-1}=A^{-1}+B^{-1}$
Find two $2\times2$ real matrices $A$ and $B$ such that $A$, $B$ , $A+B$ are all invertible with $(A+B)^{-1}=A^{-1}+B^{-1}$ Tried to write the matrices as A=\pmatrix {a&b\\c&d},B=\pmatrix {...