Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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Approximate $\mathbf{G}=(a\mathbf{H}+\mathbf{M})^+$ by Taylor expansion [closed]

Suppose we have a complex matrix $\mathbf{M}$. Let $\mathbf{M}^+=(\mathbf{M}^*\mathbf{M})^{-1}\mathbf{M}^*$ be the pseudo-inverse of $\mathbf{M}$, where $^*$ denotes the conjugate transpose. Let $\...
tam's user avatar
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Inverse of Matrix with one element approches infinity [closed]

Let A be non singular matrix of order N and inverse of A is known. Is it possible to find/approximate inverse of A if only one element of A; a(i,j); is replaced by number approaching infinity(M = big ...
mhaseeb's user avatar
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Find inverse and determinant of a symmetric matrix - for a maximum-likelihood estimation

Evaluate the determinant $\det \Omega $ and find the inverse matrix $\Omega^{-1}$ of: $$\Omega = \begin{bmatrix} \beta_1^2(1+\theta_1^2) & \beta_1 \beta_2 & ... & \beta_1 \beta_{k-1} &...
Khan's user avatar
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Inverse of a matrix expression

Let $$X_i = \left(I - P\left(I - t_it_i^T\right)\right)^{-1}$$ where $P$ is an $N\times N$ matrix and $t_i$ is a vector of $N$ elements. Is there a way to simplify this expression in order to ...
Marios V's user avatar
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Partitioned inverse 3x3 block matrix

We know that matrices can be inverted blockwise by using the following analytic inversion formula: \begin{equation} \begin{bmatrix} \mathbf{A} & \mathbf{C^T} \\ \mathbf{C} & \mathbf{D} \end{...
Cha's user avatar
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Partial inverse of a matrix - or does it have its own name?

In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name. That is, a matrix (here ...
Piotr Migdal's user avatar
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Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases. My intention is to: decompose $\Phi$ via SVD: $U \Lambda V^T = \Phi$ to create it's ...
Greyman's user avatar
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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} \...
Alec Jacobson's user avatar
4 votes
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Numerical 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 ...
sachinruk's user avatar
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optimization of inverse matrix with constraint on matrix elements

everyone! I have this optimization problem with constraint. $D$ and $T$ are symmetric matrices, where T is known and D is the unknown parameter. $x$ and $v$ are two known p-dimensional vectors. The ...
Bing's user avatar
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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 ...
user47575's user avatar
3 votes
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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 &...
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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 ...
Mathieu Galtier's user avatar
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Inverting a matrix with entries equal to positive or negative infinity

I would like to define an inverse on matrices whose entries may be positive or negative infinity. To formulate my problem precisely, suppose that I have a matrix $A$ and another matrix $B$. How do I ...
MrX's user avatar
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18 votes
4 answers
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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 Sherman-...
Federico Magallanez's user avatar
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Update the inverse of sum of two symmetric matrices

There are two invertible symmetric matrices A and B, of which B is a block diagonal. A and B have the same dimensions. I need to iteratively calculate the inverse of M = s * A + B, where s is a ...
webappl's user avatar
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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 ...
van abel's user avatar
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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 ...
Smidstrup's user avatar
6 votes
1 answer
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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} \|...
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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} \\\ \...
Dmytro's user avatar
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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 ...
Martin's user avatar
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3 votes
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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 ...
Peter's user avatar
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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 ...
Wei's user avatar
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1 answer
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Derivative of log determinant and inverse

I have a matrix $\Sigma$ with element $(i,j)$ $$\Sigma_{i,j}= \exp(-h_{i,j}\rho).$$ The matrix is positive definite and symmetric (it is a covariance matrix). Now I need to evaluate $$\frac{\...
niandra's user avatar
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8 votes
2 answers
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On the convexity of element-wise norm 1 of the inverse

Question first asked on math.stackexchange here: https://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 ...
Ferpect's user avatar
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17 votes
1 answer
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Geometric interpretations of matrix inverses

$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (...
user21816's user avatar
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bounds on the entries of an inverse circulant matrix

Suppose that $C$ is a (real) circulant invertible matrix defined by a vector $d$. Then $C^{-1}$ is also a circulant defined by some vector $f$. There exists a standard formula that expresses the ...
Felix Goldberg's user avatar
6 votes
0 answers
2k 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 ...
Felix Goldberg's user avatar
2 votes
0 answers
350 views

Fast inversion of a special kind of matrices - approximations are ok

Suppose I have a stochastic matrix $M$ (with thousands or millions of stochastic column vectors), which I split into two matrices: $D$ containing only the diagonal entries of $M$, and $R$ containing ...
Michelle Dolly's user avatar
11 votes
2 answers
9k views

How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?

I have a matrix $K$ which is the sum of a identity and a Kronecker product of two symmetric matrices as following and I want to calculate the inverse of it $K^{-1}$. \begin{eqnarray} K=\mathbf{I}_{mn}+...
ppyang's user avatar
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4 answers
604 views

inverse-closed matrix spaces

Is there a known characterization of such spaces? An example: the space of $n \times n$ matrices spanned by $I$ and $J$ (the identity and all-ones matrices, respectively) is inverse closed by the ...
Felix Goldberg's user avatar
1 vote
1 answer
852 views

sign-flipping inverse

Consider this matrix: $Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$ Its inverse is entrywise negative (you can check...) and ...
Felix Goldberg's user avatar
126 votes
13 answers
26k views

Should the formula for the inverse of a 2x2 matrix be obvious?

As every MO user knows, and can easily prove, the inverse of the matrix $\begin{pmatrix} a & b \\\ c & d \end{pmatrix}$ is $\dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{...
Frank Thorne's user avatar
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2 votes
1 answer
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Inverse of an AR(1) or Laplacian (?) or Kac-Murdock-Szegö matrix

My current problem involves having an exact (symbolic) inverse of a scaled AR(1) matrix for $n$-dimension. (I don't know what this matrix would be called in general; I'm sure it is used often.) This ...
Memming's user avatar
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15 votes
3 answers
6k views

Questions on Toeplitz matrices: invertibility, determinant, positive-definiteness

These questions are probably very basic but I'll dare to ask them anyway since I didn't have much luck in Math Stack Exchange. Let $A$ be an $n \times n$ Hermitian Toeplitz matrix: $$A = \begin{...
ght's user avatar
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2 votes
2 answers
4k views

Moore-Penrose pseudo inverse

I have an $n\times p$ matrix $Z$ with $p>n$ I have $A$, a diagonal matrix with positive entries I would like to know if there is a known relation (as a function of $A$) between the Moore-Penrose ...
Liliana's user avatar
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26 votes
6 answers
13k views

Deriving inverse of Hilbert matrix

The Hilbert matrix is the square matrix given by $$H_{ij}=\frac{1}{i+j-1}$$ Wikipedia states that its inverse is given by $$(H^{-1})_{ij} = (-1)^{i+j}(i+j-1) {{n+i-1}\choose{n-j}}{{n+j-1}\choose{n-...
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