Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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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),\...
Tatin's user avatar
  • 895
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266 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_{...
Matrix9994's user avatar
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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{...
lrnv's user avatar
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7 votes
1 answer
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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?...
cgmil's user avatar
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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 &...
Hamed's user avatar
  • 593
3 votes
2 answers
365 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 ...
Mare's user avatar
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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 $n$ ...
user0410's user avatar
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224 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 &...
Minji Kim's user avatar
2 votes
2 answers
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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.
fengbiqian's user avatar
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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 ...
fengbiqian's user avatar
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271 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: $$ \textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma) $$ ...
Soumitra's user avatar
3 votes
1 answer
378 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 < ||...
MJagota's user avatar
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2 answers
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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 ...
Viktor B's user avatar
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1 answer
779 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}...
Goulifet's user avatar
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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\|...
B Merlot's user avatar
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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 ...
Manfred Weis's user avatar
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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 $V_{n\times 1}$, $F_{m\...
sentry5588's user avatar
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$(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 $\{B_n\}_{n>1}$, $B_n\in\mathbb{R}^{m\times n}$, ...
Ludwig's user avatar
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7 votes
1 answer
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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 ...
Ranveer Singh's user avatar
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1 answer
277 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 ...
Ubo Chow's user avatar
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301 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 ...
Joshua Meggitt's user avatar
2 votes
3 answers
525 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 ...
John. Tang's user avatar
4 votes
1 answer
2k 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 $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
Christo's user avatar
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11 votes
2 answers
2k 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 ...
John Smith's user avatar
2 votes
1 answer
1k 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 ...
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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 $...
Fan Ki's user avatar
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3 votes
0 answers
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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 ...
quantum's user avatar
  • 31
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1 answer
759 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 ...
Mike Chen's user avatar
11 votes
0 answers
303 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,...
Ozzy's user avatar
  • 383
9 votes
2 answers
874 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)\...
dff's user avatar
  • 230
3 votes
1 answer
321 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 ...
mystupid_acct's user avatar
2 votes
2 answers
991 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} ...
Balaji sb's user avatar
  • 187
9 votes
1 answer
580 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 ...
valle's user avatar
  • 864
6 votes
1 answer
1k 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 $M$...
Nathan's user avatar
  • 61
4 votes
2 answers
1k 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 ...
Tarrare's user avatar
  • 143
5 votes
1 answer
193 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 ...
Dings's user avatar
  • 153
10 votes
3 answers
819 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, $$...
Dings's user avatar
  • 153
5 votes
2 answers
304 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: ...
T. Amdeberhan's user avatar
0 votes
1 answer
2k 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 $...
ffar's user avatar
  • 21
2 votes
0 answers
126 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 ...
ABIM's user avatar
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5 votes
1 answer
267 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 ...
User001's user avatar
3 votes
1 answer
2k 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 ...
Dudi Frid's user avatar
  • 255
1 vote
2 answers
1k 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}&....
Paul's user avatar
  • 41
1 vote
1 answer
165 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-...
bittertea's user avatar
1 vote
0 answers
138 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 $...
R Doe's user avatar
  • 19
1 vote
1 answer
168 views

Why polynomial $\psi^\top(t) A^{-1} \psi(t)$ attains maximum on $[-1, 1]$ at $t = \pm 1$, where $\psi_k(t) = t^k$?

Problem. Let $\psi(t) = (1, t, t^2, \ldots, t^{p-1})^\top$ - a polynomial basis. Suppose there is a matrix $$ A = \int_{-1}^1 \psi(t) \psi^\top(t) dt, \ \text{i.e. } \ A_{ij} = [2 \, | \, i+j] \...
Sergey Dovgal's user avatar
1 vote
1 answer
2k views

Largest element in inverse of a positive definite symmetric matrix [closed]

If I have an $n \times n$ positive definite symmetric matrix $A$, with eigenvalues $\lambda_{1}>\lambda_{2}\cdots>\lambda_{n}$, can I claim that the highest value which matrix $A^{-1}$ can have ...
Rohit Shukla's user avatar
2 votes
1 answer
11k views

Computing the inverse of a Cholesky decomposition [closed]

I have chol(A) and I would like chol(A^-1). One way to do this is to construct the inverse positive definite symmetric matrix and then take its Cholesky decomposition (with Dpotri and Dpotrf for ...
btracey's user avatar
  • 31
7 votes
2 answers
11k views

Relation between eigenvalues of $A$ and $A^TA$?

For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$? I ask this because I am looking into the relation between $A$ and $A+cI$, ...
Eric S.'s user avatar
  • 171
8 votes
1 answer
1k views

Efficiently compute the trace of a sparse matrix times the inverse of a sparse matrix?

How can I efficiently compute $\mathrm{trace}(A(B^{-1}))$ where $A$ and $B$ are both sparse symmetric PSD $n \times n$ matrices, both with $O(n)$ non-zero entries? If it helps, the pattern of non-...
Jeff's user avatar
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