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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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Bypassing inverse matrix calculation and the comparison of Gradient Descent and Newton Descent

I am currently reading Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John. In the 5th chapter the Gradient Descent algorithm is introduced with the following notations : $...
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Distribution of the Sum of an AR(1) Model Time Series

I have the following model for my model $\Delta X_{t} = \mu \Delta t + \rho \Delta X_{t-1} + \sigma \sqrt{\Delta t} Z_t$ with the following initial conditions - $\Delta X_{1} = \mu \Delta t + \...
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Finding inverse of $X'X$ in the case of two regressors [duplicate]

Variance of OLS etimator in matrix form look like this: $Var(\hat{\beta_j})=\sigma^2(X'X^{-1})$ I'm struggling to derive inverse matrix for the case with two independent variables. $X'X$ $=$ $\...
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numerically stable sparse gasussian process regression (matrix inversion)

In sparse approximations of GP for large data set $(X,\mathbf{y})$ with $n$ samples, usually $m$ inducing points are chosen such that the true covariance matrix is approximated by $K_{nn}\to K_{nm}K_{...
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Incremental solution for matrix inverse using Shermann-Morrison in $O(n^2)$

I have been reading a presentation on Value Function Approximation by David Silver (Introduction to Reinforcement Learning Course). On page 43 he finds a solution for linear least squares for an ...
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Which is more numerically stable for OLS: pinv vs QR

If I am doing standard OLS and want to calculate beta values (OLS estimators), which of the following is the more numerically stable method? And why? Assuming that the columns of $X$ are already mean-...
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Use $X^{-1}Y$ instead of $(𝑋^T𝑋)^{−1}𝑋^T Y$ to calculate $\beta$ when $X$ is already a square matrix in the least square problem

In the least squares problem $X\beta = Y$, the solution is $\hat{\beta} = (𝑋^T𝑋)^{−1}𝑋^TY$. I learned that two facts: $𝑋^T𝑋$ is square matrix so that the definition of matrix inversion is ...
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Proof of Pearson-Aitken selection formula

I am trying to understand the proof of the Pearson-Aitken selection formula, widely used in statistical genetics. A proof that the formula is general is given by Aitken (1936). However, I failed to ...
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49 views

Why the representation in the form of $Z'X(X'X)^{-1}X'Z$ can not be simplified into $Z'Z$

Representation similar to $Z'X(X'X)^{-1}X'Z$ frequently appear to e.g. 2SLS. I think that $Z'X(X'X)^{-1}X'Z = Z'XX^{-1}X'^{-1}X'Z = Z'(XX^{-1})(X'^{-1}X')Z = Z'Z$. So why it seems that in the context ...
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singular within scatter matrix and non singular total scatter matrix

When is the scatter matrix in linear discriminant analysis singular although total scatter matrix is non singular ? on which conditions this happens? Or can you introduce me a book or paper to read ...
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Overcoming the problem with matrices in estimating model parameters for disaggregation model

Loucks et al. (1981) defines the basic disaggregation models as ${\bf X_y = AZ_y + BV_y } $ where ${\bf Z_y} = (Z^1_y, ...,Z^N_y)^T $ is the vector of $N$ transformed normally distributed annual ...
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matrix inequality related to finance

I'm trying to show that, for certain investment strategies, it pays to have more precise estimates of the covariance matrix of your returns. I have always took this for granted, but I've been having ...
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156 views

How are biases updated when 'batch size' > 1?

This is my network represented in matrices: (a dot represents an arbitrary number) Feed-forwarding: (I omitted nesting it all in an activation function for the sake of brevity) Backpropagation The ...
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Matrices: system that is “computationally singular” versus “exactly singular” [closed]

I would like to know the mathematical concepts behind singular matrices. Matrices that do not have inverses in R throw one of two errors. I have provided some examples of both errors below: Error in ...
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Explanation of generalization of Newton's Method for multiple dimensions

I've been following the CS 229 lecture videos for machine learning, and in lecture 4 (~14:00), Ng explains Newton's Method for optimization to maximize an objective function ($f$), but doesn't clearly ...
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Recursively expressing matrix inverse

Let $X$ be an $D \times N$ matrix. Let $I$ be a $D \times D$ identity matrix. Also let $y$ be a $N \times 1$ column vector. Suppose we are trying to solve $(X X ^T + k I) w = Xy$ for a $D$ dimensional ...
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How do we know $X'X$ is nonsingular in OLS?

I am currently working through understanding the mechanics of OLS estimates and the hat matrix. One thing I have been searching for without luck is how we know that the term $X'X$ is invertible where $...
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Intuition behind $(X^TX)^{-1}$ in closed form of w in Linear Regression

The closed form of w in Linear regression can be written as $\hat{w}=(X^TX)^{-1}X^Ty$ How can we intuitively explain the role of $(X^TX)^{-1}$ in this equation?
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Why does higher variance of independent variables decrease standard errors of the estimator?

Some time ago, reading Advanced Data Analysis from an Elementary Point of View by Cosma Rohilla Shalizi, I found this statement in a section about significant coefficients: Moreover, at a fixed ...
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Use of Moore-Penrose Inverse in Likelihood Computation

I've been running MLE on a mixed-Gaussian model (let's just focus on the multivariate Gaussian case for now). In order to take the inverse of the covariance matrix, I've been using MATLAB's pinv() ...
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48 views

Newton-Raphson Error

According to Agresti(2013) pg 364-365, iterative methods such as Newton-Raphson methods, $ \begin{aligned} \beta^\text{new} &= \beta^\text{old} + (X^{T}WX)^{-1}X^{T}(V) \end{aligned} $ help to ...
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Are the Inverses of two asymptotically equivalent matrices themselves asymptotically equivalent

Suppose $M_n = P_n + op(1)$. Is it the case that $M_n^{-1} = P_n^{-1} + op(1)$, if both $M_n^{-1}$ and $P_n^{-1}$ exist with probability going to 1 as $n$ increases? Can the Continuous Mapping ...
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156 views

Obtaining the possible least squares solutions when $X^TX$ is not invertible

If $X^TX$ is not invertible, what is the set of solutions for the least squares estimator $\hat{\beta_1}$ in the below? $Y_i = \beta_0 +\beta_1(x_i-\bar{x}) +\epsilon_i$ I got as far as writing out ...
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156 views

“Undefined real result” with inverse() function in WinBUGS

Using WinBUGS, I'd like to generate the posterior distribution for a function of several other variables (stochastic nodes). One step in building up this function involves taking the inverse of a ...
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Expectation of a Matrix Raised to a Power

I have been able to find little literature related to the topic online (e.g., A note on the Expected Value of an Inverse Matrix such that E($\frac{1}{X})$ $\geq$ ($\frac{1}{E(X)}$), though, vaguely, I ...
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Least Square Solution to Linear Regression Problem

Given two column-vectors, A = [0; 1.6818; 2.8284; 3.8337; 4.7568; 5.6234] and B = [0; 984.7; 1590.7; 2029.1; 2251.9; 2254.45], I ...
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Section on Moore-Penrose Pseudoinverse in the Deep Learning book

I am reading the following section in the Deep Learning book by Goodfellow, Bengio and Courville. I have some questions. If $A\in\mathbb{R}^{m\times n}$ and $Ax=y$, then we can have left inverse ...
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Why can't we cancel these two matrices in the OLS estimator?

When I have a multivariable linear regression model for sample element $i$: $$y_i=\beta_1x_{1,i}+...+\beta_kx_{k,i}+\epsilon_i,$$ then the OLS estimator is determined by the following equation: $$X^Ty=...
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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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628 views

The probability limit of an inverse matrix

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 ...
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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 ...
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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 ...
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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 ...
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1answer
101 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 ...
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407 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 & ...
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1answer
229 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 $C$ ...
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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_{...
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when can I substitute an inverse with a pseudo-inverse in an estimator

Short Version: can I substitute the Moore-Penrose generalized inverse of a matrix (R function ginv()) for a matrix inverse (R function ...
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84 views

Generalised inverses - Solve normal equations

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: ...
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861 views

Efficient/feasible sparse matrix inversion in R

I am looking to perform a 2-stage least-squares estimation with sparse matrices in R, in the style of Bramoulle et al (J. Econometrics 2009). Specifically, let: G be a very sparse block-diagonal ...
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611 views

Solving simple linear equation using Matrices [closed]

I need to find a matrix A whose dimensions will be 1 x n and I have input matrix X whose ...
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1answer
478 views

When is it appropriate to override the default reciprocal condition number tolerance for solve() in R?

I am estimating a GMM IV model, where I'm creating a weighting matrix by taking the inverse of Z'Z, where Z is a matrix of instruments. For certain combinations of instruments, when I try to compute ...
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What is the physical significance of inverse of a matrix? [closed]

I was asked this question in an interview. Though I tried my best to answer the question in whatever way I could (I was explaining in terms of mathematics), the professor looked upset. Any idea? The ...
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2answers
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Residual Sum of squares in Weighted regression

The Residual Sum of squares (RSS) in Weighted regression is written as $$(\mathbf{y-X\hat{\boldsymbol\beta}})^{'}\mathbf{C}^{-1}(\mathbf{y-X\hat{\boldsymbol\beta}})$$ Where $$\hat{\boldsymbol\beta}=(\...
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How to anticipate bad solutions to system of equations?

I know the identity that matrix $\mathbf{Y}$ is a product of $\mathbf{X}$ and $\mathbf{B}$ (all of which are integer matrices in my case): \begin{equation} \mathbf{Y} = \mathbf{X} \mathbf{B} \end{...
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What is an example of perfect multicollinearity?

What is an example of perfect collinearity in terms of the design matrix $X$? I would like an example where $\hat \beta = (X'X)^{-1}X'Y$ can't be estimated because $(X'X)$ is not invertible.
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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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255 views

Evaluate high-dimensional Gaussian with variance matrix $\sigma^{2}I_{n_{t}\times n_{t}}+\boldsymbol{\Sigma}_{t}\boldsymbol{\Sigma}_{t}^{'}$

I need to compute the log-likelihood function in a high-dimensional Gaussian time-series. I have the following model: $\mathbf{y}_{t}\left|\mathcal{F}_{t-1}\sim\mathcal{N}\left(\mathbf{\boldsymbol{\...
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The Existance of Schur Complement Inverse

A block matrix $\mathbf{M}=\left[ \begin{array}{ccc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^T & \mathbf{C} \end{array} \right]$ is invertible if $\mathbf{A}$ and $(\mathbf{C}-\mathbf{B}^T\...