Linked Questions

0
votes
1answer
341 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....
0
votes
1answer
49 views

Effective way to calculate the inverse (A+kB)^-1 with k changing and A, B fixed

I have a Simulink modell where I need to calculate $(A+c_k B)^{-1}$ in every time step with $c_k$ changing each iteration. Does someone know any more effective way to do it, instead of calculating a ...
0
votes
1answer
29 views

Inverse of $S_{N,N-1}'AS_{N,N-1}$ where $S$ is orthogonal and $A$ is of full rank.

Let $S\in\mathbb{R}^{N\times N }$ be an orthogonal matrix and denote $S_{N,N-1}\in\mathbb{R}^{N \times N-1}$ as the matrix with the same elements of $S$ but without the last column of $S$. Let $A\in\...
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' \\ \...
-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 ...
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
vote
0answers
27 views

Reduced complexity of matrix inversion of sum of rank 1 matrices: $M^{-1} = \left( I + \sum \limits_{i=1}^{K} \alpha_i A_i \right)^{-1} $?

How to reduce the complexity of matrix inversion of sum of rank 1 matrices, not only arithmetic but also run time? $M^{-1} = \left( I + \sum \limits_{i=1}^{K} \alpha_i A_i \right)^{-1} $ where $A_i ...
1
vote
0answers
132 views

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 ...
1
vote
0answers
362 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 ...
1
vote
0answers
592 views

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 ...
0
votes
0answers
26 views

Convexity of a function over matrices [duplicate]

How to show that $f(X)=tr(X^{-1})$ is convex over the set of symmetric real positive definite matrices by showing that $g(t)=f(X+tV)$ is convex for any such $X$ and $V$? Clearly we need to show that $$...
0
votes
0answers
16 views

Simplification of $x = \left(A + \sum_i \alpha_i B_i\right)^{-1} A \ y$, $A \succ 0, A \in M_{n,n}(\mathbb{C})$, $\textrm{rank}(B_i) = 1$

I have a matrix inversion problem on hand. I want to reduce the matrix inversion complexity, if at all feasible. Let me give a brief overview of the problem definition. Problem definition: Say $A \...
0
votes
0answers
34 views

How to calculate $(I + GH)^{-1}$? [duplicate]

Suppose we have a matrix A. We have decomposed A into a sum of the identity matrix and the product of two column(G) and row(H) matrices. In general, how can we calculate $A^{-1}$?
0
votes
0answers
47 views

Matrix expression manipulation

Given the left hand side of the equation below, how to show the equality with the right hand side? I checked it numerically, but not sure how to prove it. \begin{align} A - AC^T(CAC^T +R)^{-1} CA = (...
-2
votes
0answers
36 views

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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