Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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22 views

Confusion about the invertible condition of a banded and bordered matrix

Recently, I noticed a conclusion in the textbook "Handbook of Splines" on pages 307-308, please see screenshots as follows: Here I give an example with order $n=5$ I tried to send an e-...
3
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0answers
117 views

Minimum upper bound for sum of the entries of the inverse covariance matrix

Let $x \in \mathbb{R}^n$ and $k$ is RBF kernel $$k(x, x') := \exp \left(-\frac{\|x-x'\|^2}{2\sigma^2}\right)$$ and let $\mathbf{K}$ be the following $n \times n$ covariance matrix $$\mathbf{K} = \...
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1answer
125 views

If $\Vert A-B\Vert_{op}\leq \varepsilon$ then $A^{-1}$ and $B^{-1}$ are uniformly close

Let $A,B$ are two $p\times p$ positive definite matrices such that $0<\delta_0\leq \min\{\lambda_{\min}(A), \lambda_{\min}(B)\}\leq \max\{\lambda_{\max}(A), \lambda_{\max}(B)\}\leq \delta_1$. Also ...
4
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2answers
190 views

Do any two hermitian matrices A and B commute with the support of their commutator?

Let $A$ and $B$ be Hermitian matrices. Let $[A,B] = AB - BA$ be their commutator and let $[A,B]^+$ be the Moore-Penrose pseudoinverse of $[A,B]$. Is it then true that $A$ and $B$ both commute with the ...
2
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1answer
375 views

Mistake in Karl Pearson's 1900 paper introducing the chi-squared distribution

Background I'm reading Karl Pearson's 1900 paper titled: On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be ...
2
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1answer
86 views

Matrix derivative w.r.t. a general inverse form: $(A^TA)^{-1/2}D(A^TA)^{-1/2}$

I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have $$\frac{\partial}{\partial A_{ij}} a^T(A^TA)^{-1/2}...
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2answers
97 views

Computation to differentiate a determinant [closed]

Consider a positive Hermitian $N \times N$ matrix $A$ with complex valued coefficients. We list its eigenvalues in increasing order and with their multiplicities, $\mu_{1} \leq \mu_{2} \leq \cdots \...
1
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1answer
186 views

Ratios of Gaussian integrals with a positive semidefinite matrix

Cross-post from MSE https://math.stackexchange.com/questions/4118128/ratios-of-gaussian-integrals-with-a-positive-semidefinite-matrix Generally speaking, I’m wondering what the usual identities for ...
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2answers
353 views

Sufficient conditions for invertibility of a block tridiagonal matrix

Let $M_n \in \mathbb{R}^{N \times N}$ be a block-tridiagonal matrix: $$M_n = \begin{bmatrix} B_1 & C_1 & 0 & 0 & \cdots & 0 \\ A_1 & B_2 & C_2 & 0 & \cdots & 0 \...
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59 views

Algorithm for computing group inverse of some matrices

I am interested in algorithms (and implementations) for computing the group inverse of $\rho(A)I - A$ of a non-negative irreducible matrix $A$. In the well-cited reference [1], the author describes in ...
2
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1answer
80 views

Closed form for the inverse of a special transition matrix

Let $b\in\mathbf{R}^n$ be fixed, and let $A\in\mathcal{M}_{n\times n}(\mathbf{R})$ be given but otherwise arbitrary. Let $a_1,\ldots,a_n$ denote the coefficients of the characteristic polynomial of $A$...
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1answer
62 views

Expected pseudo-inverse of isotropic random matrix

Suppose I have a random $m \times m$ matrix $R \sim \mu$ that is possibly singular. Is it true that $ E[R] \propto I$ implies that there exists a scalar $r_{\mu, m}$ such that $E[R^+] = r_{\mu, m}I$, ...
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23 views

Approximate distribution for leverages of the ridge hat matrix in a randomised framework

For a couple of days, I have been struggling to extend some results of the distribution of the leverages for random Hat matrices to the case where we add a penalizing term. The setting is the ...
3
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1answer
88 views

Condition number for matrix of eigenvectors of a diagonalizable matrix

Let $A$ be a diagonalizable matrix, i.e., $A=SDS^{−1}$. For any matrix $A$, condition number is defined as $\kappa(A)=\|A\| \|A\|^{-1}$. For $A$ being a diagonalizable matrix, define $G_A=\{{S: S^{-1} ...
2
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1answer
125 views

Matrix inversion inequality

Suppose $A, B, C\in\mathbb{R}^{n\times n}$ are all symmetric positive definite matrices, and they satisfy the inequality $A \succeq B + C$. Assume also that all of the three matrices are bounded, i.e.,...
2
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1answer
99 views

Complexity of pseudoinverse of a low-rank matrix

I have a large matrix $\mathbf{A}\in \mathbb{C}^{m\times n}$ with very low rank $r$, I find that the general complexity of finding its pseudoinverse is $\mathcal{O}(\max(m,n)^3)$, this is too high. ...
32
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1answer
3k views

Inverting lower triangular matrix in time $n^2$

I have a lower $n\times n$ triangular matrix called $A$ and I want to get $A^{-1}$ solved in $O(n^2)$. How can I do it? I tried using a method called "forward substitution", but the ...
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110 views

How to calculate inverse of sum of two Kronecker products with specific form efficiently?

I have a matrix with specific form of $A\otimes I + B\otimes J$ where $A$ and $B$ are general dense matrices, $n\times n$. $I$ is an $m\times m$ identity matrix. $J$ is a $m \times m$ dense matrix ...
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83 views

Random matrix invertible

I am trying to figure out why the following random matrix is invertible: \begin{align*} A_j = I_d + J_{\mu}(\hat{X}_{t_{j-1}})(t_j - t_{j-1}) + \left( \begin{array}{rrr} B_1^T \\ ...
5
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1answer
240 views

Inverse of a Cauchy-like matrix

Consider $n\times n$ symmetric Cauchy-like matrix $M$ with elements $(M_{ij})_{i,j=1}^{n}$ given by $$M_{ij} = \frac{1}{(n-i)!(n-j)!(2n-i-j+1)} = \displaystyle\int_{0}^{1}\frac{x^{n-i}}{(n-i)!} \frac{...
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0answers
58 views

Fast computation of linear equation with row and column removed

Given $A, x, b$ and a linear system $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ is full rank. Denote $A_{\backslash i}$ as a $(n-1)\times(n-1)$ matrix where $A$ removes its $i^{th}$ column and row, ...
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1answer
183 views

Inverse of a larger matrix where the inverse of the submatrix is known

Let $A, A^{-1} \in \mathbb{R}^{n \times n}$ be known matrices. Suppose we have an invertible matrix $B \in \mathbb{R}^{(n+1) \times (n+1)}$ of the following form: $$B = \begin{bmatrix} A & ...
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1answer
102 views

Low-complexity method for sub-matrix inversion

Assume that $\mathbf{A}$ be an $N\times N$ matrix. We know that the complexity of the computation of matrix inversion is $\mathcal{O}(N^3)$. Let define $\mathbf{D}=\mathbf{A}^{-1}$. Now, assume that $\...
3
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58 views

Regularity of Moore-Penrose pseudo-inverse

Let $k\in\mathbb{N}\cup\{0\}$, let $\Omega\subseteq\mathbb{R}^n$ be open, connected and let $G\in C^k(\Omega;\mathbb{R}^{n\times n})$ satisfy $$ \operatorname*{rank}G(x)= \operatorname*{rank}G(y),\...
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78 views

Bound for the inverse of a summation of rank-1 matrices

Given vectors $x_1,\dots,x_T \in R^d$ satisfying $\|x_i\|_2 = 1$, define $A_0 = I$ and $A_t = I + \sum_{i=1}^tx_ix_i^\top$ for $t \geq 1$. We are interested in the following quantity: \begin{align} S_{...
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0answers
94 views

Is this matrix invertible?

Suppose that $P$ is a finite partition of the unit $d$-dimensional hypercube composed of $m$ hyperrectangles $R_1,...,R_m$ Denote $u_1,...,u_m$ the middle-point of each hyperrectangle : $u_i = \frac{...
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1answer
1k views

What are known properties of matrices where off-diagonal elements are 1?

Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties?...
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51 views

Solutions to a special confluent Vandermonde system

Consider the polynomial $P(X) =\prod_{i=1}^k (X-x_i)^s$ and let $M$ be the corresponding confluent Vandermonde matrix. Concretely, here is what I mean by that. Define $$ M^{(0)} = \begin{pmatrix} 1 &...
3
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2answers
310 views

Inequality for 0-1 matrices

Given an $n \times n$-matrix $A$ with entries only $0$ or $1$ and determinant equal to $\pm 1$. Define the magnitude $M_A$ of $A$ as the sum of all entries of the inverse of $A$. Question 0: Do we ...
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1answer
231 views

A closed-form expression for the inverse of a block-matrix

Let $\bf A$ be an $n \times n$ non-singular matrix over $\mathbb{F}$. Let $x$ be a non-zero element of $\mathbb{F}$. Suppose that ${\bf 1}_{n}$ is a symbol for the all-one vector of length $n$ ...
3
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93 views

Singularity of symmetric block matrix with singular diagonal blocks

One can show that the following statement holds: Given symmetric matrix $A \in \Re^{n \times n}$ and tall matrix $B \in \Re^{n \times p}$ with full column rank, $$\begin{bmatrix}A & B \\ B^T &...
2
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1answer
476 views

How to derive the mean of inverse-Wishart distribution?

How to derive the mean of inverse-Wishart distribution in Inverse-Wishart distribution? I have no idea about it. Thanks for your help.
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67 views

What is the expected inverse of 1 plus a Wishart distribution?

Let $$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$ Where $\mathcal{W}$ denotes the Wishart distribution and $\Sigma \in \mathbb{R}^{q \times q}$ is the corresponding scale matrix (symmetric, positive ...
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0answers
178 views

Direct solution to maximum likelihood computation problem using the derivative of multivariate Gaussian w.r.t. covariance matrix

For an application, I need to compute the maximum loglikelihood of data coming from a $d$-dimensional multivariate Gaussian random variable: $$ \textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma) $$ ...
3
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1answer
133 views

Difference of pseudoinverse bound under assumptions

This seems like a standard problem, but unable to find a solution online. Suppose we have two singular PSD matrices A and B with the following assumptions: $ 0 < x \leq ||A|| \leq y$ $ 0 < ||...
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2answers
115 views

Controllability Gramian asymptotics for small times

Set-up. Consider the following linear controlled system $$ \dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1) $$ where $y$ is the state of the system, $y(t) \in R ^n$, $A \in R ...
6
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1answer
369 views

Trace of inverse of random positive-definite matrix in high dimension?

Consider a random matrix $A \in \mathbb{R}^{n\times n}$ with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of $$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
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0answers
87 views

Angle inequality for inverse of PD diagonal matrix

I have two diagonal, positive definite matrices $A$ and $B$, and I have that the angle between $Ax$ and $x$ is smaller than that of $Bx$ and $x$: $$\frac{x^T A x}{\|Ax\|_2} \geq \frac{x^T B x}{\|Bx\|...
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27 views

Explicit Formula for the Pseudo Inverses of a Family of Matrices Related to Calculating Certain Vertex Weights

Trying to identify canonical vertex weights for complete weighted graphs, I investigated the Ansatz, that the sum over the weights of all edges that are adjacent to same vertex, should equal the sum ...
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357 views

How to solve optimization with matrix pseudo inverse

Thank everybody in advance. I'd like to solve an optimization problem for a matrix function $f(C)$. However, the matrix pseudo-inverse constraint gives me big troubles. Vectors $V_{n\times 1}$, $F_{m\...
5
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1answer
338 views

$(AB)^+\approx B^+A^+$ for $B$ "fat" enough?

Let $A\in\mathbb{R}^{r\times m}$ be a matrix of full row rank, and let $\cdot^+$ denote the Moore-Penrose inverse. Consider a sequence of matrices $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
7
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1answer
1k views

Determinant and Inverse of a Toeplitz matrix

Let $T(n,k)$ be a $n \times n$ symmetric Toeplitz matrix, where all the entries of first $k$ super-diagonal (and sub-diagonal), last $k-1$ super-diagonal (and sub-diagonal) are ones, and rest of the ...
0
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1answer
101 views

How to derive the solution of Tikhonov Regularization via SVD [closed]

The solution to Tikhonov Regularization is $$x=(A^HA+\sigma^2_{min}I)^{-1}A^Hb$$ where $\sigma^2_{min}$ is the minimum of the singular values of $A$. Then we apply $SVD$ to $A$ such that, $$A=U\Sigma ...
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0answers
194 views

Derivative of complex matrix pseudo inverse with respect to real and imaginary components

I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$. I am interested in evaluating the ...
2
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3answers
362 views

Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently? $$(D + ADA^T)^{-1}$$ Hints or references are ...
3
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1answer
1k views

Approximating the expectation of a matrix inverse

Let $$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$ where $A$ is a given $n \times m$ matrix (where $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
11
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2answers
1k views

Inverse of a small submatrix

Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
2
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1answer
828 views

Inversion of the sum of an identity matrix and two Kronecker products

Writing the SVDs of symmetric positive-definite matrices $A=U_AD_AU_A^T$ and $B=U_BD_BU_B^T$, the following inversion can conveniently be simplified: \begin{align*} (I + A \otimes B)^{-1} &= ((U_A ...
3
votes
2answers
3k views

Inverse of particular lower triangular matrix

I have an $n \times n$ lower triangular matrix $A$ where $$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$ $$A_{ii}=1, \quad 1 \leq i \leq n,$$ and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...
3
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0answers
182 views

The inverse of sum of two positive matrices with almost orthogonal supports

I am interested to find an approximate formula for $$A (A+B)^{-1} A\ ,$$ for two positive matrices $A$ and $B$ whose supports are almost orthogonal. If the support of $A$ and $B$ are orthogonal ...