# 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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### Why is least squares failing on my exact linear equation? (sanity checking) [closed]

I have a problem with 3 free variables. Right now I am in a simulation phase, so I can create an arbitrary number of example cases for regression. This is a linear problem, so I can write my data in ...
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### How to overcome the issue of singular within-cluster scatter matrices in clustering using entropy-based feature ranking?

I am trying to implement the entropy-based feature selection method for clustering by Dash and Liu. In this method, features are ranked in importance based on an entropy minimisation procedure and ...
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### Squared Multiple Correlation (SMC) of my correlation matrix tend towards 1. How to interpret this?

In order to be able to conduct exploratory factor analysis, I want to carry out parallel analysis to determine the number of factors to be extracted. To do so, I want to extract the eignevalues of the ...
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### Upper bound trace of inverse of covariance matrix

Let C be the covariance matrix from any normal distribution. If the trace of C is upper-bounded by a constant k (i.e., tr(C)<=k), can I find an upper bound for the trace of the inverse of C (i.e., ...
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### Generalized Least Squares using Moore Penrose pseudo inverse

I'm using GLS to fit a model where some independent variables are strongly correlated. Therefore my covariance matrix is singular. I have found that Moore-Penrose pseudo inverse can be used to find an ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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}=(\...