40 questions linked to/from Inverse of the sum of matrices
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### How to find the inverse of sum of matrices [duplicate]

If I have matrix A and matrix B how do I find the inverse of (A+B), the sum of the matrices and A is invertible but B isn’t
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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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### 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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### Real parts of diagonal elements of $(I-U)^{-1}(I+U)$ are zero?

I noticed numerically that real parts of diagonal elements of $(I-U)^{-1}(I+U)$ are zero (assuming $I-U$ is invertible), where $I$ is identity matrix and $U$ is a unitary matrix ($U^\dagger U=I$). A ...
### If $A^{-1}$ has been precomputed, is there an efficient way to compute: $(A+λI)^{-1}$
If $A^{-1}$ has been precomputed (or to be more precise: the Cholesky decomposition of A has been precomputed and cached), is there an efficient way to compute either $$C = (A+λI)^{-1}$$ or (more ...
This question is very much related to Inverse of the sum of matrices . Let $d\ge 1$ be an integer and let $a\in(-1,1)$ and $b\in(-1,1)$ be real numbers. Let \${\bf C}:=\left( f(|i-j|) \right)_{i,j=1}^{...