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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Invertibility of the Gram matrix of a convex combination

Let's assume two real valued matrices $A,B\in R^{w\times d}$ for which $d>w$ and they both have full (column) rank. I am interested in the invertibility of the Gram matrix $$H(t):=(A+t(B-A))(A+t(B-...
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inverse of canonical correlation

The canonical correlation obtains the most correlative pairs of components across two matrices. However, the inverse is not commonly known but could be useful for obtaining the most different ...
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How to obtain least squares when $X^TX$ cannot be inverted

This work is all theoretical and for school, so we were only provided this information to work with, no actual y values. I have a simple linear model I have been asked to translate into a matrix, ...
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how can an inverse correlation matrix have <1 in diagonals. the correlations x1, x2 are about 0.5 [duplicate]

The inverse correlation matrix was calculated with the minverse function. The results show that in the diagonal the values are less than 1. how is this possible? the x1,x2 independent variables are ...
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Inverse of the outer product of some vectors with their transpose

Assume i have $n$ 3D unit vectors $v_s$, with different values. Then i define a matrix $T$ as: $$ T = \frac{1}{n} \sum_{s=1}^{n} v_s \times v_s' $$ where $v_s$ are $3\times 1$ vectors and therefore $...
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Positive definiteness of integral of matrix

I was reading a paper, and did not understand a statement that the author made without further explanation. The author derives the limiting distribution of a non-linear least-squares estimator and ...
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Condition for covariance matrix to be non-invertible

Context: I'm working on a machine learning problem where I'm using multivariate normal likelihood which requires calculating determinant and inverting the covariance matrix. I'm trying to generate ...
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How to derive an inverse of Gaussian Kernel

As an example, say I have a function (Gaussian process kernel): $$K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)+\beta \delta_{ij}$$ Is there a way to analytically express $K^{-1}(x_i,x_j)$, s.t. ...
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How to prove $(P^{-1} + B^T R^{-1} B)^{-1} B^T R^{-1} = PB^T(BPB^T + R)^{-1}$

It is Equation C.5 from https://www.seas.upenn.edu/~cis520/papers/bishop_appendix_C.pdf I tried right multiply both sides with $(BPB^T + R)$, but not sure how to continue from there.
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How can nuisance parameters in Fisher matrix can deteriorate the useful constraints?

I have a Fisher matrix $F$ which has the matrix blocks form like this : $$ F=\begin{bmatrix} A & B\\ C & D \end{bmatrix} $$ The block $A$ is the most important block, in the sense the ...
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Problem dealing with OLS estimator

I'm an econometrics student and I'm having a little trouble with lineal algebra. I have seen that the OLS estimator, given the following regression in matrix form: $$ y=X \beta+u $$ Is: $$ \hat{\beta}=...
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How to simplify fallowing martrices' Expected value elements in the equation

I have a matrix equation with four variables inside, $x^1_{00}$, $x^1_{tt}$ and $x^2_{tt}$, $x^2_{tt}$. $x^1_{00}$, $x^1_{tt}$ come from the same distribution , they are only shifted by timelag $t$. ...
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Distribution of solution to linear system

I have a random symmetric matrix $ A \in \mathbb{R}^{M \times M}$, and random vector $b \in \mathbb{R}^M$. I also have access to expressions for the mean and variance of each element of $A$ and $b$ (...
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Distribution of matrix products

Let's say I have two random vectors $A, B \in \mathbb{R}^N$ that are distributed approximately normally, with distribution $\mathcal{N}(\mu, \Sigma)$. I then define $\bar{A}$ as being a diagonal ...
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Correlation matrix with 0 determinant [duplicate]

I am looking at crypto coin price data. I compute the correlation matrix but I am unable to invert due to a zero determinant. I'm not quite sure why this is happening as none of the columns are that ...
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When is the Optimal weighting matrix in GMM singular?

currently I am trying to estimate a simple linear regression: \begin{equation} y_t = X \beta + \varepsilon_t, \end{equation} where I try to find 4 coefficients and one specific predictor is an ...
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Singularity of a modelmatrix that consists of one predictor raised to multiple powers

I was playing around a little bit with linear regression models in R and wanted to try out whether I can get a perfect fit for a linear model where I have a response $$Y = \left( \begin{array}{c} y_1\...
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Mercer's theorem and eigenfunctions

Consider a kernel $K$ which satisfies the conditions of Mercer's theorem. We know that $K(x,y) = \sum\limits_{i=1}^{+\infty}\lambda_ie_i(x)e_i(y)$ where $\lambda_i$ and $e_i$ are the eigenvalues/...
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Finding inverse matrix $(X'X)^{-1}$ with $X$ as design matrix [duplicate]

I'm relatively new to all this and I am trying to figure out how I can derive the matrix $(X'X)^{-1}$ when I have given $x_1, x_2, x_3$ and $y$. $X$ is the design matrix in that case but not sure how ...
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linear regression and obfuscation matrix - just not clicking for me?

Here is the basis of my question. We have an exercise to take a data set of made up insurance claims and train a linear regression model to predict future claims; however, because it is sensitive data,...
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completing a square

If I have a density function of the form $p(x) \propto \exp(−q(x)/2)$ where $q(x)$ has the following quadratic function $$q(x)=x^Tx+y^Ty-[x^TA+y^TB][A^TA+B^TB+\beta\mathbb{I}]^{-1}[A^Tx+B^Ty]$$ where $...
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What is an example of a systems problem that can be solved by either 'matrix inversion' or an 'iterative procedure' to arrive at the same result?

I recently heard over a radio program (in French) that a given problem - can't recall it exactly, but it involved solving large systems - could be solved "through matrix inversion" or "...
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Relationship between Cholesky decomposition and matrix inversion?

I've been reviewing Gaussian Processes and, from what I can tell, there's some debate whether the "covariance matrix" (returned by the kernel), which needs to be inverted, should be done so ...
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What dimensionality reduction methods allow a lower dimensional reconstruction of the original data besides PCA via invertible transformations?

In eigenfaces, one used the inverse transformation PCA is capable of doing to reconstruct the low dimensional face image. In tsne one may not reconstruct the original dataset to produce something akin ...
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Why doesn't $(e^{A})^{-1} = e^{-A}$ hold for a symmetric matrix in Python?

$e^A$ is just the $A$ matrix with all of its elements exponentiated, called a matrix exponential. It follows that the inverse $(e^{A})^{-1} = e^{-A}$ for square matrices, although I could find nothing ...
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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 ...
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4 votes
1 answer
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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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6 votes
2 answers
589 views

Least Squares removing first $k$ observations Woodbury formula?

Given the matrix $X_{n,p}$ from the least squares problem $$ \mathbf{X} \cdot \mathbf{\beta} = z $$ Where the normal equation is: $$ \mathbf{\hat{\beta}} = \left(\mathbf{X}^T \mathbf{X}\right)^{-1} ...
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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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1 vote
1 answer
342 views

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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Fast inverse of positive definite matrix subtracted by its Nystrom approximation

Assume a positive definite symmetric covariance matrix $$C_{n,n}$$ and let its Nystrom approximation be $$\hat{C}_{n,n}=C_{n,q} C^{-1}_{q,q} C_{q,n}$$ for some $q<n$ Inverting $C_{n,n}$ is of $\...
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5 votes
1 answer
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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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1 vote
1 answer
495 views

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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2 votes
0 answers
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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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3 votes
1 answer
484 views

numerically stable sparse Gaussian 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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6 votes
1 answer
475 views

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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6 votes
1 answer
2k views

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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3 votes
1 answer
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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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1 answer
156 views

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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3 votes
1 answer
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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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1 vote
1 answer
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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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1 vote
1 answer
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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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3 votes
1 answer
4k views

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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3 votes
1 answer
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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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