# Questions tagged [matrix-inverse]

For questions about inverses and pseudoinverses of matrices.

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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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### 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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### 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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### 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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### 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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### 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: $$\textbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$$ ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...