# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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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 ...
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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 ...
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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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### 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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### 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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### 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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### 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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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, $$\begin{bmatrix}A & B \\ B^T &... 2 votes 1 answer 778 views ### How to derive the mean of inverse-Wishart distribution? How to derive the mean of inverse-Wishart distribution in Inverse-Wishart distribution? I have no idea about it. Thanks for your help. • 171 1 vote 0 answers 78 views ### What is the expected inverse of 1 plus a Wishart distribution? Let$$ X \sim \mathcal{W}_{n} (n, \Sigma) \; \;$$Where \mathcal{W} denotes the Wishart distribution and \Sigma \in \mathbb{R}^{q \times q} is the corresponding scale matrix (symmetric, positive ... • 171 1 vote 0 answers 248 views ### Direct solution to maximum likelihood computation problem using the derivative of multivariate Gaussian w.r.t. covariance matrix For an application, I need to compute the maximum loglikelihood of data coming from a d-dimensional multivariate Gaussian random variable:$$ \textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma) $$... • 11 3 votes 1 answer 281 views ### Difference of pseudoinverse bound under assumptions This seems like a standard problem, but unable to find a solution online. Suppose we have two singular PSD matrices A and B with the following assumptions:  0 < x \leq ||A|| \leq y  0 < ||... • 33 1 vote 2 answers 131 views ### Controllability Gramian asymptotics for small times Set-up. Consider the following linear controlled system$$ \dot{y}(t) = A y(t) + B u (t), \ \ t \in [0,T], \ \ \ \ \ \ \ \ \ \ (1) $$where y is the state of the system, y(t) \in R ^n, A \in R ... • 664 7 votes 1 answer 589 views ### Trace of inverse of random positive-definite matrix in high dimension? Consider a random matrix A \in \mathbb{R}^{n\times n} with i.i.d. entries, with symmetric law and finite variance. I am curious about the behavior of$$\mathrm{Tr}( (A^T A + \lambda \mathrm{Id})^{-1}...
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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\|...