For questions about inverses and pseudoinverses of matrices.

learn more… | top users | synonyms

2
votes
0answers
18 views

Efficiently factorize a KKT system with block diagonal upper corner

I have a system resulting from a quadratic energy minimization with linear equality constraints enforced with Lagrange multipliers which has the form: \begin{equation} A = \left[\begin{array}{c|c} ...
1
vote
0answers
24 views

Trace of Inverse matrix from Cholesky

This question was somewhat answered here: Fast trace of inverse of a square matrix. However, I feel like there was no complete answer wrt the Cholesky case. I have the matrix $\Sigma=LL^T$. Is there ...
0
votes
1answer
24 views

optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint. D and T are symmetric matrice, where T is known and D is the unknown parameter. x and v are two known p-dimensional vectors. The objective ...
1
vote
2answers
85 views

Bound on smallest entry of inverse matrix

For a symmetric, invertible matrix $A=(a_{ij})\in \mathbb{R}^{n\times n}$ with (at least two) nonzero off-diagonal elements, is it possible to bound in absolute value the smallest entry of its inverse ...
3
votes
1answer
146 views

Calculating a generalized inverse (Moore–Penrose pseudoinverse)

I am considering a graph with $n$ edges with the following nicely structured adjacency matrix: \begin{equation} A_n= \begin{pmatrix} 0 & 0 & 0 &\cdots & 0 & 0 & 1\\ 0 & 0 ...
0
votes
1answer
85 views

Perturbation of Cholesky decomposition for matrix inversion

I am looking for a computationally cheap way to compute $x$ such that $$(L L^T + \mu^2 I)x = y$$ where $L \in \mathbb{R}^{n \times n}$ is a lower triangular definite positive matrix (with some very ...
4
votes
3answers
201 views

Sherman-Morrison type formula for Moore-Penrose Pseudoinverse

Given an $n\times n$ invertible matrix $\mathbf A$ and two column vectors $\mathbf u$, $\mathbf v\in\mathbb R^n$, suppose that $1 + {\mathbf v}^T {\mathbf A}^{-1}\mathbf u \neq 0$. Then the ...
1
vote
1answer
86 views

How to calculate $(A^{-1})_{ii}$ for an invertible hyperhermitian quaternionic matrix $A$?

The article Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238. has the following CLAIM: Claim. Let $A$ be an invertible hyperhermitian ...
1
vote
1answer
111 views

Do there exist a way to solve inhomogeneous matrix equations Ax = b for only selected rows?

The inhomogeneous matrix equation $\mathbf{A} x = b$ can be solve in many ways, but in this particular case, I am looking for a solution to this problem on a set of constraints. The matrix $A$ is ...
5
votes
0answers
223 views

How to calculate the inverse of a matrix with block Kronecker structure?

I have a Hermitian matrix $A$ with the following block Kronecker structure, $A = \begin{pmatrix} R_0 \bigotimes S_0 & R_1 \bigotimes S_1\\ R_1^H \bigotimes S_1^H & R_0 \bigotimes S_0 ...
6
votes
1answer
244 views

Best rank one approximation

Assume $u,v\in\mathbb{C}^n$ are complex vectors. I was wondering if there is a closed form expression for the following problem in terms of $u$ and $v$ \begin{equation*} \arg\min_{x\in\mathbb{C}^n} ...
1
vote
0answers
85 views

positiveness of the inverse solution to Sylvester equation

I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form: $$ \mathbf{M} = \begin{vmatrix} \mathbf{A} & \mathbf{b} \\\ ...
0
votes
0answers
286 views

Diagonal of the inverse of a 6x6 symmetric partitioned matrix

Let $$M = \begin{bmatrix} A & B \\ B & C \end{bmatrix}$$ in which $A$, $B$ and $C$ are $3 \times 3$ matrices being also symmetric. In fact, they are quite similar, just differing on a single ...
3
votes
1answer
308 views

Expected value of trace of matrix inverse

Given a $N\times K$ matrix $A$ of full rank with $ K < N $, a diagonal matrix $D$ and knowing that $E[D]=bI_N$, where $E[\cdot]$ is the expected value and $I_N$ is the $N\times N$ identity matrix ...
0
votes
1answer
243 views

Calculate the inverse of a matrix

Hi I have a equation $$Ax=b,$$ where matrix $A$ is invertible, $b$ is a constant vector, and $x$ is the unknown vector. To obtain $x$, it is obvious $x=A^{-1}b$. Alternatively, if $A$ is Hurwitz, one ...
6
votes
2answers
128 views

On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: http://math.stackexchange.com/questions/317209/on-the-convexity-of-element-wise-norm-1-of-the-inverse On the convexity of element-wise norm 1 of the ...
5
votes
0answers
391 views

When is the inverse diagonally dominant?

There is a large literature devoted to studying the inverses of diagonally dominant matrices. I'd like to know if there is information about a so-to-say opposite situation: we have a matrix $A$ and ...