# Tagged Questions

The inverse of a given square matrix, $A$, is the matrix $A^{-1}$ such that $AA^{-1}$ is the identity matrix.

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### Inverse of block covariance matrix

I have a positive definite symmetric covariance matrix which looks like this: A, B, C, D and E, F, G are MATRICES, also positive definite symmetric covariance What is the inverse of such a matrix? ...
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### Efficient routines for a regression with orthogonal regressors?

I have a standard OLS regression setup, where (sets of) the regresors are orthogonal to each other. I am looking for a fast low-level way (using qr() instead of <...
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### How to calculate the inverse of sum of a Kronecker product and a diagonal matrix

I want to calculate the inverse of a matrix of the form $S = (A\otimes B+C)$, where $A$ and $B$ are symetric and invertible, $C$ is a diagonal matrix with positive elements. Basically if the ...
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### Why does inversion of a covariance matrix yield partial correlations between random variables?

I heard that partial correlations between random variables can be found by inverting the covariance matrix and taking appropriate cells from such resulting precision matrix (this fact is mentioned in ...
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### Invertibility of $X^TX$ with severe multicollinearity in regression

I am studying about multicollinearity in regression and in the book it says, "if there is severe (but not perfect) multicollinearity, two or more predictor variables are highly correlated, so $X^TX$ ...
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### Matrix Inversion Error

I a Multiple linear regression model, from published literature, I am implementing a spreadsheet to generate new predictions based on the published model. the literature stated Coefficients and the ...
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### R - system is computationally singular - dealing with small numbers

I'm working with a ~200x200 Markovian transition matrix of non-zero probabilities. Forcibly, these probabilities are, for the large part, going to be very small. I am trying to find the inverse of my ...
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### Condition Number in Single Variable Regression

I've calculated a single variable linear regression using OLS (in the python statsmodels library). The model has a large condition number, and suggests there may ...
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### Is it posible to find a derivative of the mean function of Gaussian process regression?

The mean function $\hat{\mu}(x_*)$ of GPR is $k(x_*, X)(k(X, X) + \sigma^2_w I)^{-1}Y$ where $k(\cdot, \cdot)$ is a kernel matrix or vector of appropriate size and is parametrized by some ...
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### Diagonal elements of the inverted correlation matrix

Is it true that the diagonal elements of the inverted correlation matrix will always be larger than 1? Why?
I want to find the inverse of the following matrix: $$R_{k-1}=\begin{pmatrix} 1 &\rho &\rho^2 &\cdots &\rho^{k-2} \\ \rho &1 &\rho &\cdots &\... 1answer 474 views ### How to proof relationship between inverse covariance matrix and linear regression coefficients? Edited: I would like to work out the above relationship, more precisely: Let (Y_{1}, ..., Y_{m}) be a zero-mean vector with covariance matrix \Sigma, and let S \subset \{1, ..., m\}. The ... 0answers 83 views ### A non-negative definate matrix has a non-negative generalized inverse I'm having trouble proving a N.N.D matrix has a N.N.D G-Inverse. So far I have: If we assume x = Az where x >= 0 and A is a nnd matrix. So if Y is a G-inverse than: x = Az = YAz = Yx >= 0 . Thus ... 1answer 143 views ### Orthogonalizing predictors for least squares estimation I know that orthogonalization in LS is to avoid inverting X'X. The idea behind it is to find variables Z that are orthogonal to each other. Although the process to find those is clear to me, I don't ... 1answer 276 views ### Tolerance for pseudoinverse In calculating the pseudoinverse of a matrix A, of size (m,n), I need to choose a tolerance threshold for the eigenvalues. I'm trying to understand how I should pick this. Matlab default is to use ... 2answers 473 views ### Inverting non positive definite covariance matrix I have an expression for a covariance matrix C in terms of the indices i and j. In this way I can analytically calculate the elements of my covariance matrix, however when I try to invert C ... 2answers 199 views ### Returning the inverse of a matrix in a quadratic program (SVM) in cvx optimization package I am solving the dual QP of an SVM, and using the RBF kernel. As you know, the objective function is of the form$$f(\alpha) = \alpha^T Q \alpha $$where \alpha is the optimization variable and Q ... 1answer 106 views ### Uniqueness of x'\beta even when \mathbb{E}(x^Tx) is not invertible As discussed in user25658's answer to this question, when one wants to compute$$ \beta = \mathbb{E}(x^Tx)^{-1} \mathbb{E}(x^TY) $$but \mathbb{E}(x^Tx) is not invertible, \beta is not uniquely ... 0answers 126 views ### Interpretation of regression coefficients obtained from applying left inverse of regressor matrix in an underdetermined system? If X^\dagger is the pseudo-inverse of X, \beta = X^\dagger y is the least squares solution for \beta when y=X\beta. In the overdetermined case, applying X^{\dagger,L} = (X^TX)^{-1}X^T ... 2answers 129 views ### IV estimator: efficient implementation? I would like to implement (in R) an instrumental variable (IV) estimator, that takes the most general form (here not 2SLS or GMM!):$$ \beta_{IV} = (Z'X)^{-1}Z'Y  I could code this in the naive way,...
I'm curious about the nature of $\Sigma^{-1}$. Can anybody tell something intuitive about "What does $\Sigma^{-1}$ say about data?" Edit: Thanks for replies After taking some fantastic courses, I'd ...