# Questions tagged [matrix-inverse]

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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### Proving that the hat matrix is unchanged even when the predictors are multiplied by constants

I know that the hat matrix $H = X(X^T X)^{-1} X^T$, and that $\hat{Y} = HY$. When we have some non-zero constants that we multiply each respective predictor by, which just multiplies every column in ...
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### Why is it not possible to simplify $b=(X'X)^{-1}X'y$ [duplicate]

Why is the following not possible: $$b=(X'X)^{-1}X'y = X^{-1}(X')^{-1}X'y=X^{-1}y$$ While this term $(AB)^{-1}=B^{-1}A^{-1}$ applies to any two matrices as long as both are of full rank and are $nxn$...
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### Moore Penrose Pseudo-Inverse Fast Algorithm in R [closed]

I want to apply Moore Penrose Pseudo-Inverse on my matrix, which is a 20,000 * 20,000 symmetric matrix with rank 19,999. I found ginv() function from the ...
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### How to do multiple linear regressions with overlapping predictors efficiently

I'm using covariance matrices to do a linear regression. I have the same number of predictors and dependent variables (3000 of each). For each $i$, $X_{i}$ and $Y_{i}$ have a kind of symmetry. I ...
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### NN type/architecture needed for inverse covariance matrix approximation

The idea is to construct a neural network (NN) that takes N series of financial returns as input and returns the (approximation of the) inverse of the sample covariance matrix (an N times N matrix). I ...
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### Normalization for solving linear equations

Suppose I want to solve a linear equation system in the form of $$A x = b$$ to get $x$, where $A$ are $n$ by $n$ matrix and $b$ is $n$ by 1 vector. Is there any normalization procedure necessary ...
167 views

### Updating the covariance matrix after deleting the i-th column and row

Suppose I have a covariance matrix $A_{n\times n}$ and $A^{-1}_{n\times n}$ is its inverse. Then I randomly exclude the $i$-th row and column, where $1\le i\le n \in \mathbb{N}$, obtaining the new ...
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### Why forward and reverse transforms for PPCA are so different?

In probabilistic PCA (PPCA), they model dimensionality reduction as probabilistic model $t = Wx + \mu + \epsilon$ where $W$ is non-swuare matrix of size $(d \times q)$, $q < d$ $x$ is a vector ...
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### Under-constrained models and invertibility of covariance matrix

In Goodfellow et al.'s Deep Learning, the authors write on page 232: [$\mathbf{X^\top X}$] can be singular whenever the data-generating distribution truly has no variance in some direction, or when ...
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all: I am considering a question regarding the calculation of the probability limit for an inverse matrix. Specifically, suppose we have a non-singular and squared matrix $M$ with dimension $2\times ... 1answer 768 views ### Usefulness of convexity of linear regression when there is no closed form solution The optimisation problem in linear regression,$f(\beta) = ||y-X\beta||^2$is convex (as it is a quadratic function), and when$(X^TX)$is invertible, we have a unique solution which we can calculate ... 1answer 2k views ### Non-Singularity due to inclusion of non-zero lambda in ridge regression [duplicate] There were many similar questions on this site , related to this but none were exactly to the point I wanted to ask So the question is relates to ridge regression and This link where there is a ... 0answers 343 views ### Solving an inverse problem with machine learning I am running up against a very tough inverse problem that I suspect might be solvable using machine learning. Here is the problem. Overview I am studying an object$X$which, internally, is ... 1answer 146 views ### Invert singular matrix for design of experiments and regression I have a matrix,$X'X$, which is singular meaning that I cannot invert it. I need the inverse of this matrix to perform two independent things. I need it for the design of experiments, in R using ... 1answer 618 views ### How to show that demeaning the data in design matrix does not change the hat matrix When I have a design matrix $$X = \begin{bmatrix} 1 & x_{11} & \ldots & x_{1k}\\ 1 & x_{21} & \ldots & x_{2k}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & ... 1answer 279 views ### Spectral Decomposition of a symmetric matrix times a diagonal matrix I would like to find the inverse of the sum of a Kronecker product and a diagonal matrix. I found this answered here, but I don't see how the last step is valid. Because of the fact that C ... 0answers 523 views ### How to compare diagonal elements of precision matrix (the inverted correlation matrix)? Let$$C=\begin{pmatrix}C_{11} & C_{12}\\ C_{21} & C_{22}\end{pmatrix}$$be a$p\times p$correlation matrix with positive entries, where$C_{11}$is a$q\times q$matrix. Define$D=C^{-1}=(d_{...
I am still not fully able to handle the concept of generalised inverses when applied to OLS. Is there a way to show that $\beta$ = $\ (X'X)^- X'y + (I_k−(X'X)^-X'X)z$ solves the normal equations: ...