Linked Questions

2
votes
1answer
213 views

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 ...
4
votes
3answers
76 views

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 {...
1
vote
2answers
150 views

Inverting a special matrix

Consider matrices $A$ and $B$ of the forms below: $$A = \lambda \cdot I$$ $$B = \beta \cdot \pmatrix{ 1 & 1 & \cdots & 1\\ 1 & 1 & \cdots & 1\\ \vdots & \vdots & \...
3
votes
1answer
239 views

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. ...
0
votes
1answer
463 views

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 ...
0
votes
1answer
342 views

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....
1
vote
0answers
364 views

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 ...
4
votes
1answer
128 views

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 ...
1
vote
3answers
118 views

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?...
0
votes
1answer
61 views

Inverse of the $(A + iB)$

I have taken this question from Zhang, Fuzhen. Linear Algebra (Johns Hopkins Studies in the Mathematical Sciences) Assuming that all matrix inverses involved below exist, show that $(A + iB)^{...
1
vote
1answer
93 views

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 ...
1
vote
1answer
122 views

Symbolic inverse of a linear combination of two matrices

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}^{...
0
votes
1answer
95 views

Representation of the inverse of an variance-covariance matrix $\hat{\Sigma}^{-1}$

Given $T$ observed vectors $x_i\in\mathbb{R}^N, i\in\{1,\ldots,T\}$. Define $\hat{\Sigma}$ as the corresponding empirical covariance-matrix of the Observations $X=\left(\begin{array}{c} x_1' \\ \...
4
votes
0answers
70 views

Computation of Matrix inversion

Suppose $\mathbf{A}\in \mathbb{R}^{n\times n}$ is non-singular and its inverse $\mathbf{A}^{-1}$ is known. We would like to compute $\mathbf{B}^{-1}$ where we assume that $\mathbf{B} = \mathbf{A}+\...
-1
votes
1answer
53 views

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 ...

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