# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

122
questions

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### Algorithm for efficiently calculating $(A+\sum_{i=1}^n B_i)^{-1}$ where $A^{-1}\in\mathbb S^n_+$ is known and $B_i$ are sparse matrices

Let $A\in\mathbb R^{n\times n}$ be a symmetric positive-definite matrix and $A^{-1}$ is already known. Now I want to compute the matrix $(A+\sum_{i=1}^n B_i)^{-1}$ where each $B_i$ is a sparse ...

2
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2
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### Orthonormal solution of overdetermined linear equations

I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that:
$$AX = B$$
Given that $X$ is ...

2
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2
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178
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### Full-rank matrix

I have a sparse square matrix and want to see if it is full rank (so that I can apply the implicit function theorem).
$$\left[\begin{array}{cccccccccc}
0 & 1 & 1 & 1 & 0 & 0 & ...

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### Recover the parameters of an affine transformation matrix

I am using a 3D Affine Transformation matrix of the form $T(x) = RGS (x-c) + t + c$, with:
$R= R_x R_y R_z$ the rotation matrix,
$G$ the shearing matrix,
$S$ the scaling matrix,
$c$ the center of ...

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1
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### Approximating the expectation of trace inverse of random Gaussian combination

Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. Has anyone studied this expectation in asymptotics $$E_{A}(\mathrm{Tr}( (A^T ...

2
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1
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86
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### Trace inverse of random PSD matrix?

Consider a random matrix $A \in \mathbb{R}^{m\times n}$ with i.i.d. entries, with mean zero and variance 1 and $m <n $. I am interested in the expectation of $$E_{A}(\mathrm{Tr}( (A^T A + \lambda \...

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### Proving 2 matrices have the same trace [closed]

I found a problem in my textbook and I have tried solving it, but I had no succes. The problem is:
Let $A$ and $B$ be $n \times n$ matrices with complex number entries. Given that $AB−BA$ is ...

1
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1
answer

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### Characterization of pseudo unit-matrices

I accidently found a class of rank-deficient square matrices that are their own Moore-Penrose pseudo inverse:
$$\boldsymbol{A}\in\mathbb{R}^{n\times n}: a_{(i,i)}=n-1,\, a_{\lbrace i,j\rbrace}=-1\...

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86
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### Fast inverse of asymmetric diagonally dominant matrix with diagonal 1 and non-positive off-diagonals

I am interested in ways to obtain (even approximately) the inverse of an asymmetric diagonally dominant matrix with diagonal 1 and non-positive off-diagonals.
Formally, let $A$ be a $n\times n$ matrix ...

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### Condition on the probabilities for the $J\times J$ matrix $[ \Pr(X=j \mid Y=k) ]$ to be invertible

$\DeclareMathOperator\Pr{P}\newcommand\cPr[2]{\Pr(#1 \mid #2)}$I have a $J \times J$ matrix:
$$
M:= \begin{bmatrix}
\cPr{X=1}{Y=1} & \cPr{X=2}{Y = 1} & \cdots & \cPr{X=J}{Y = 1} \\
\cPr{X=...

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1
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### Does there exist a function $f(X)$ with the following gradient $\mathrm{Tr}[(I-X)^{-1}]\cdot g(X)$?

Let $f: \mathbb{R}^{n\times n} \to \mathbb{R}$ be a function that receives a square matrix and spits out a scalar. Does there exist a function $f$ such that the gradient $\nabla_Xf(X) = \mathrm{Tr}[(...

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### Product of matrices equal identity

I need to solve the following equation for the matrix $P \in\mathbb{R}^{r\times d}:$
$$
((PAP^\top)^{-1} P S P^\top (PAP^\top)^{-1})^2 = I_r,
$$
where $S$ is a symmetric $d\times d$ matrix, $A$ is a ...

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0
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49
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### Derive an inverse of Gaussian process kernel (or a parametrized matrix)

As an example, say I have a function (Gaussian process kernel):
$$K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)+\beta \delta_{ij}$$
Is there a way to analytically express $K^{-1}(x_i,x_j)$, s.t. ...

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### When is $(I - X)^{-\top} \circ X = 0$?

I am currently looking at the following expression $(I - X)^{-\top} \circ X = 0$, where $\circ$ is the Hadamard product, $\top$ is the transpose, $I$ is the identity, and $X$ is non-negative and ...

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### Existence of matrices with some invertibility properties

Prove that there exists five matrices $B_i \in \mathbb{F}_2^{5\times 10}$, $i\in \{1,2,3,4,5\}$, such that any two $B_i$'s form an invertible matrix in $\mathbb{F}_2^{10\times 10}$.
I am interested ...

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1
answer

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### 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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1
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### 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 ...

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### 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 ...

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### 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 ...

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1
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292
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### 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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2
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### 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 \...

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1
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235
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### 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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501
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### 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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### 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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1
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### 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 $E[R^+] = r_{\mu, m}I$, ...

3
votes

1
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### 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} ...

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### 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.,...

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235
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### 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. ...

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### 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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3
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302
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### 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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0
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### 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 \\ ...

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1
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314
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### 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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0
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### 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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### 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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### 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 $\...

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0
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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),\...

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### 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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0
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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{...

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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?...

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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 &...

3
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2
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329
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### 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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### 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 $n$ ...

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### 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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1
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778
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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.

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0
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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 ...

1
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0
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### 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
votes

1
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281
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### 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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2
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131
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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 ...

7
votes

1
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589
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### 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}...

1
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0
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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\|...