Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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38 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
110 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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37 views

pseudo-inverse of short fat matrix

Consider the matrix $A \in \mathbb{R}^{m \times n}$ with $m < n$. Is its pseudo-inverse $A^\dagger = (A^\top A)^{-1} A^\top$ computable ? I'd expect not, because $A^\top A$ is not of full rank ( $...
1
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1answer
77 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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55 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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63 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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89 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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59 views

Unboundedness of pseudo-inverse

Let $H$ be a real Hilbert space, and let $T$ be a bounded self-adjoint operator of $H$ to itself. Let $T^+$ denote the pseudo-inverse operator of $T$, that is the operator defined to be zero on $(TH)^{...
7
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1answer
656 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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43 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
293 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
158 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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48 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
159 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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55 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
57 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
63 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
76 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 ...
5
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1answer
231 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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83 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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0answers
22 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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197 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\...
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1answer
319 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
816 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 ...
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1answer
80 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
141 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
309 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
481 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
884 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
595 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 ...
2
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2answers
2k 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
154 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 ...
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1answer
237 views

What is the most accurate and efficient method of finding an inverse of a hessian matrix?

For any hessian matrix, of say 300 by 300, and may or may not necessarily be positive semi-definite, thus algorithms such as Cholesky decomposition may not be used. I've found that some algorithms ...
11
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0answers
263 views

Jaffard's theorem - finite matrices

For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies $$ A(k,l) \leq C (1+\left|k-l\right|)^{-r}, $$ for some $C>0$, then $$ A^{-1}(k,...
9
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2answers
658 views

Inverse of special upper triangular matrix

Consider the following $n \times n$ upper triangular matrix with a particularly nice structure: \begin{equation}\mathbf{P} = \begin{pmatrix} 1 & \beta & \alpha+\beta & \dots & (n-3)\...
3
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1answer
190 views

Inverting (via Taylor expansion) a sum of (rank-deficient) skew-symmetric matrix and (rank-deficient) Diagonal matrix

I have the following problem: A matrix $C\in \mathbb{R}^{2N}$, where $C=\epsilon A+D$ $\epsilon A=(C-C')/2$ is skew symmetric with "block" anti-diagonal structure of size 4. $ D=(C+C')/2$ (Diagonal ...
2
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1answer
418 views

When is the following block matrix invertible?

Let $$A = \begin{bmatrix} x_{11} A_{11} & x_{12} A_{12} & x_{13} A_{13} & \cdots & x_{1d} A_{1d}\\ x_{21} A_{21} & x_{22} A_{22} & x_{23} A_{23} & \cdots & x_{2d} ...
9
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1answer
338 views

Inverse of a matrix with binomial entries

This is closely related to this question: Eigenvalues of a matrix with binomial entries. We consider the matrix: $$M_{ij} = 4^{-j}\binom{2j}{i}$$ where it is understood that the binomial ...
5
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1answer
448 views

Invertibility of the Schur Complement

Suppose that $$ M = \begin{bmatrix}A & B \\ C & D\end{bmatrix}. $$ I know that if $D$ and $M\setminus D$ (where $M\setminus D$ is the Schur Complement of $D$ in $M$) are invertible, then $M$...
3
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2answers
652 views

Derivative of pseudoinverse with respect to original matrix

I have been trying to find an analytical expression for the following: $\frac{\partial {X^{+}}}{\partial {X}}$ In my case, $X$ has a constant rank. I've found the formula for differentiating a ...
5
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1answer
123 views

Find the inverse of a more general matrix that is similar to the Hilbert matrix

In the last MO question , the following matrix is given: $$M_{ij}=\left[\frac{1+(-1)^{i+j}}{i+j-1}\right]$$ and its inverse has been discussed. Now the problem is further extended to a more general ...
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3answers
667 views

Find the inverse of a matrix that is very similar to the Hilbert matrix

The standard Hilbert matrix $H$ is given by $$H_{ij}=\frac{1}{i+j-1},$$ and it has an inverse given for example in this MO question. Now I have encountered a matrix $M$ of similar form, namely, $$...
5
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2answers
253 views

minimum-maximum entries matrix

Let $M(n)$ be an $n\times n$ matrix in the variables $x_1,\dots,x_n$ with entries $$M_{i,j}(n)=\frac{x_{\max(i,j)}}{x_{\min(i,j)}}, \qquad 1\leq i,j\leq n.$$ I'm interested in the following: ...
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1answer
1k views

How to calculate the inverse of the sum of kronecker products with the identity matrix

How to calculate $G^{-1}$ efficiently when $G$ is a large matrix knowing that: \begin{eqnarray} G=I⊗A + A⊗I \end{eqnarray} Or since i'm using $G^{-1}$ to multiply by some other matrix, how to find $...
2
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0answers
124 views

Left multiplication Fréchet Differentiable

Let $L^2$ be a separable Hilbert space and let $\{e_i(t)\}_{i \in \mathbb{N}}$ be a basis for it. Moreover let $\phi(T,z)$ and $\psi(T,z)$ be 1-parameter families (in z) in $L^2$, with basis ...
3
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1answer
164 views

Are inverse eigenvalue problems (IEPs) hopeless and not a fruitful area of research?

I've been studying IEPs, in particular, the Nonnegative Inverse Eigenvalue Problem, some basic theoretical framework, the many open questions that IEPs have, and now sort of realize the computational ...
3
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1answer
1k views

Transforming a non-invertible matrix into an invertible matrix

Given a non-invertible, diagonalizable matrix $A$, I wish to transform it into another matrix $B$ that satisfies: $B$ is invertible all non-zero eigenvalues of $A$, are also eigenvalues of $B$ all of ...
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2answers
745 views

Is there an easy way to find the inverse of a symmetric block matrix with off-diagonal rank-one matrices?

The symmetric matrix I need to invert is of the following form: \begin{align} J_e=\left(\begin{matrix}-I&B_{11}&B_{12}&...&...&B_{1(N-1)}\\ B_{11}&-I&B_{22}&....
1
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1answer
147 views

Find a matrix and its inverse satisfying lower and upper bounds

I reduced a problem of matrix completion to the problem find $A,B$ such that $AB=I$ $A_{min}\leq A \leq A_{max}$ $B_{min}\leq B \leq B_{max}$ One possible approach would be to just minimize $\|AB-...
1
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0answers
119 views

Computing Moore-Penrose generalized inverse using convergent geometric series

I came across an article (published in IEEE transactions or Wiley publications) which states that moore-penrose generalized inverse can be computed using the following two equations. Let $A$ be an $...