40 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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### 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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### 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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### Inverse of sum of 3 matrices

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 ...
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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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### Show that $A^T=A^{-1}$

Say I have a matrix $B$ that is skew symmetric ($B^T=-B$). I want to show that for $A=(I+B)(I-B)^{-1}$ that $A^T=A^{-1}$ is true. All we know is that $B$ is square and that $(I-B)$ is non singular. ...
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### Concavity of trace of positive definite matrix

I have to show that $Tr((A^{-1} + B^{-1})^{-1})$ is a concave function, being A and B positive definite matrices. I cannot imagine how is this possible since we are computing the trace of a positive ...
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### harmonic mean of covariance matrices.

After writing up some math, I ended up with a term like so: $\left(A^{-1} + B^{-1}\right)^{-1}$ where $A$ and $B$ are 2 covariance matrices. 1) Can I be sure that this expression is meaningful? (i....
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### Pseudoinverse of the sum of matrices

Similarly to the question posted here Inverse of the sum of matrices but in case of non-square matrices. If I want to compute the pseudoinverse of (A+B) and matrices A,B pinv(A) is known is there a ...
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### Is there a way to update the inverse of a sum of two matrices following a rescaling of one of them?

Suppose I have two matrices $A$ and $B$ (let's assume that both $A$ and $B$ are invertible, as is their sum), and a scalar $g$. I am interested in the matrix $$M^{-1} = (A + gB)^{-1}$$ I am aware ...
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### Any lemma for $(A+A^{-1})^{-1}$?

I'm actually a little surprised since I wasn't able to find any nice property to compute $(A+A^{-1})^{-1}$ ... Anyone knows about a theoretical way to achieve this ? Like an specific inversion lemma?...
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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 ...