56 questions linked to/from Inverse of the sum of matrices
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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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### 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: $$\min \sum_{i=1}^n (y_i-a^\top x_i)^2, \quad \text{s.t } \|a\|_{2} = 1.$$ ...
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### Inverse or approximation to the inverse of a sum of block diagonal and diagonal matrix

I need to calculate $(A+B)^{-1}$, where $A$ and $B$ are two square, very sparse and very large. $A$ is block diagonal, real symmetric and positive definite, and I have access to $A^{-1}$ (which in ...
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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)+...