Questions tagged [determinant]

determinant of a matrix. For purely mathematical questions about determinant, better ask at mathSE

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Multicollinearity iff determinant correlation matrix = 0

I'm studying Linear Models again, after finishing my degree some years ago. I found in my old notes that, according to my professor, one can check multicollinearity calculating the determinant of the ...
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Generalized variance of a multivariate normal without calculating determinants

In order to calculate ‘generalized variance’ of a multivariate normal distribution, it is often recommended (e.g., here: https://online.stat.psu.edu/stat505/lesson/1/1.5) to calculate the determinant ...
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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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Impute Missing Correlations from Incomplete Correlation Matrix in R

I am in need of a function that will take an incomplete correlation matrix and return a complete matrix with correlations imputed for missing values. I am working on models that utilize only summary ...
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Is negative Mahalanobis distance proportional to log probability?

Is the following statement true? I was sure it was, but I was told by someone else that it is not. $$ p(\mathbf{x}_i | y_i) = \frac{1}{2\pi^{\frac{D}{2}} |\mathbf{\Sigma}_c|^{\frac{1}{2}} } e^{ -\frac{...
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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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If all elements of A are greater than x and all elements of B are smaller than a, det(A) is greater than det(B)

Let $A$ and $B$ are two $n \times n$ matrices and $x > 0$ is a scalar. If $\forall \; a \in A \;\;\; a > x$ and $\forall \; b \in B \;\;\; 0 \leq b \leq x$, and assume $A$ and $B$ are both ...
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Is there a formula for the determinant of the covariance matrix $\mathbf{X_n}^T \mathbf{X_n}$ in the case of multiple regression?

Consider the standard simple linear regression model: $$ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i, $$ for $i=1,\dots,n$. In matrix-vector form this is $$ \mathbf{Y} = \mathbf{X_n}\beta + \epsilon, $$ ...
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Question regarding Optimal Designs of experiments

I'm a bit unclear on the concept of optimal design of a data matrix $X$. I propose a small example to work through: Suppose $\epsilon_i \sim N(0, \sigma^2)$ are i.i.d., and I have some experiment ...
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Given a function $\phi:X \to Y$ and a target distribution $\pi_Y$ on $Y$, does there exist a distribution $\pi_X$ on $X$ s.t. $\phi(x) \sim \pi_Y$?

More formally, given $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$, $\phi: X \to Y$ and distribution $\pi_Y$ on $Y$, does there exist a distribution $\pi_X$ on $X$ such that $\phi(x) \sim \pi_Y$ ...
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Multivariate gaussian bivariate gaussian proof

I'm having trouble seeing how the multivariate gaussian formula evaluates to the bivariate gaussian. See multivariate PDF, source: http://cs229.stanford.edu/section/gaussians.pdf [![multivariate][1]][...
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What's the importance of parallel eigenvectors?

I'm studying eigenvectors. I read that if a matrix is symmetric and if the eigenvalues are real numbers, the eigenvectors will be perpendicular. However, I have no idea what it means (if anything) ...
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Is the determinant of the correlation matrix << 0.00001 necessarily a problem for PCA?

I am running PCA on my data and my KMO value > 0.80 and p-value of Bartlett's test of sphericity < 0.05. However, the determinant of the correlation matrix ( around 10^-30) is very close to zero....
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How to prove the determinant of covariance matrix is zero, when n≤p?

If there are n observations on p dimensions, then the covariance matrix will be: But when n≤p, its determinant will be zero. I know it is because it becomes as a singular matrix, but I do not know ...
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What is the meaning of $\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$?

What is the meaning of the quantity: $$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$ Is there, for example, a geometric explanation? Is there a term for it in statistics?
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Linear dependency among columns and rows

Singular matrix is defined as square matrix with the determinant of zero. The determinant of zero occurs when matrix columns are linearly dependent (i.e. one of the columns can be defined as a linear ...
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Singular Matrix and Linear Dependency

Singular matrix is defined as a square matrix with determinant of zero. I am aware that linear dependency among columns or rows leads to determinant being equal to zero (e.g. one column is a linear ...
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2 votes
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Understanding how the determinant of the multidimensional normal likelihood can overrule the prior probability

I am doing Bayesian inference. I have a normal prior probability distribution of some theoretical parameter $\theta$ and I am trying to update my knowledge of $\theta$ using some data $D$ and a model $...
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Reason for absolute value of Jacobian determinant in change-of-variable formula?

When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a ...
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Name of a second order statistical quantity similar to generalized variance

I am looking for the name of a statistical quantity similar to generalized variance in a 2 dimensional space. Specifically, I define generalized variance as $\epsilon_{a,b}^2 = \left|\begin{matrix}\...
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Why does the determinant of the Hessian grow with n?

Context: I'm trying to understand BIC on a deeper level. I'm using BIC for model/structure selection for Bayesian networks. I'm confused because BIC is an approximation to the likelihood of a model, ...
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How to model for independent determinants in several groups based on follow-up time

I want to answer the research question which determinants are associated with long-term survival after a myocardial infarction (MI) in a prospective patient cohort study. More precisely: I want to ...
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Do the Determinants of Covariance and Correlation Matrices and/or Their Inverses Have Useful Interpretations?

While learning to calculate covariance and correlation matrices and their inverses in VB and T-SQL a few years ago, I learned that the various entries have interesting properties that can make them ...
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Multicollinearity (or not) in exploratory factor analysis

I’m performing an exploratory factor analysis with 28 items, n = 300. I’m confused whether I have a multicollinearity problem or not, and if so whether/how I go about choosing items to remove from the ...
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Algebraic derivation of canonical correlations

In a paper from 1936, Harold Hotelling (access on JSTOR) defined the concepts of canonical correlations and canonical variates for two sets of variates. In pages 327 and 328, he precisely derives ...
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Maximum determinant with $l_2$ norm penalties

Let the random sample $Y_1,\ldots,Y_n\sim N_p(0,\Sigma)$, then the likelihood is given by, \begin{align*} L(\Sigma)=\frac{1}{(2\pi)^{np/2}|\Sigma|^{n/2}} e^{-\frac{1}{2}\sum_i Y'_i\Sigma^{-1}Y_i} \end{...
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Log Expectation of Inflated Determinant of Wishart Distribution

Let $\Lambda \sim \mathcal W(\nu, \Psi)$, i.e., following a $n \times n$ dimensional Wishart distribution with mean $\nu \Psi$ and degrees of freedom $\nu$. The expectation of the log determinant of $\...
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Determinant of a block matrix with sparse elements

I have a positive definite symmetric matrix that looks like $$\pmatrix{A & 0 & 0 & E \\ 0 & B & 0 & F \\ 0 & 0 & C & G \\ E^\prime & F^\prime & G^\prime &...
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How to get the determinant of a covariance matrix from its diagonal elements

I am trying to implement a speaker recognition system in MATLAB. I am using Gaussian Mixture Models (GMM) for speaker modelling and maximizing the posterior probabilities for classification. The ...
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Stability of VAR$(p)$: Prove $\det(I_{Kp}-\tilde{A}z)=\det(I_k-A_1z-\ldots-A_pz^p)$

Given we have a VAR$(p)$ process written in the companion form $$\tilde{y}=\tilde{v}+\tilde{A}\tilde{y}_{t-1}+\tilde{u}_t$$ where $$\tilde{A}=\left(\begin{array}{ccccc} A_1& A_2 & \ldots &...
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How is the determinant of $(X'X)$ related to variance?

I'm working on a problem (and actually have the answer) but I don't know why this is the answer, can someone explain this equality?. It has to do with the the determinant of the partitioned matrix $(X'...
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1 answer
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What's special about (x'x-x'Proj(w)x)?

I'm working on my homework and I keep seeing something like $$(x'x-x'W(W'W)^{-1}W'x)$$ I know that $W(W'W)^{-1}W'$ is the projection matrix, but what is so special about subtracting those two inner ...
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2 votes
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Column1 = Temp_F and Column2 = Temp_C -- are these linearly dependent?

If $X$ is a matrix of size $m$ x $2$, where the first column is a range of Celsius values and the second column is their corresponding Fahrenheit values, would the columns be considered linearly ...
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Meaning of determinant of matrix of impulse-response functions in a VAR

Suppose there is a structural VAR model, such as: $A y_{t} = B y_{t-1} + \varepsilon_{t}$, where $\varepsilon_{t} \sim N(0, I_{n})$; then the matrix representing the contemporaneous impulse-response ...
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1 answer
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Sampling from an (almost) multivariate normal over matrices

Consider $n$ points in the euclidean plane, $p_i = (x_i,y_i)_{1\leq i \leq n}$. Now consider a $2 \times 2$ matrix $M = \left(\begin{array}{cc}a & b\\c& d\end{array}\right)$ a vector $r = \...
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Applying inferential statistics for census data

Let's assume I have a census data of a population which I would like to study and it has variables such as age, gender, sex, occupation etc and the dependent variable which is community participation ...
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How to generate uniformly random orthogonal matrices of positive determinant?

I've got probably a silly question about which, I must confess, I'm confused. Imagine repeated generating of uniformly distributed random orthogonal (orthonormal) matrix of some size $p$. Sometimes ...
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10 votes
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Fisher information matrix determinant for an overparameterized model

Consider a Bernoulli random variable $X\in\{0,1\}$ with parameter $\theta$ (probability of success). The likelihood function and Fisher information (a $1 \times 1$ matrix) are: $$ \begin{align} \...
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1 vote
1 answer
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Determinant of the covariance matrix in a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol \...
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9 votes
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Why people often optimize the determinant of $(X'\Sigma X)^{-1}$

Say I have a random vector $Y\sim N(X\beta,\Sigma)$ and $\Sigma\neq\sigma^2 I$. That is, the elements of $Y$ (given $X\beta$) are correlated. The natural estimator of $\beta$ is $(X'\Sigma^{-1}X)^{-1}...
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3 votes
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What does Determinant of Covariance Matrix give?

I am representing my 3d data in covariance matrix. I just want to know what the determinant of a covariance matrix gives. If the determinant is positive, zero, negative, high positive, high negative, ...
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1 answer
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How to calculate Eigenvalue without using eign() function with R? [closed]

I understand how to calculate EIGN by hand but when I try to write code without function EIGN(), I did not have a clue. To calculate Eigenvalue is to count all the possible c in ...
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9 votes
3 answers
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Why do we use the determinant of the covariance matrix when using the multivariate normal?

I am not well versed in statistics. I wanted to know why we use the determinant of the covariance matrix instead of having the covariance matrix itself when writing down the multivariate normal ...
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53 votes
3 answers
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Why does correlation matrix need to be positive semi-definite and what does it mean to be or not to be positive semi-definite?

I have been researching the meaning of positive semi-definite property of correlation or covariance matrices. I am looking for any information on Definition of positive semi-definiteness; Its ...
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4 votes
1 answer
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Factor analysis: What to do when determinant is almost zero and when KMO for a variable is low?

I'm conducting a factor analysis on 40 interval-level variables, and have two immediate concerns: The determinant is 6.608E-006, which is much lower than the cut-...
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MLE estimation of spatial effects radius

I am trying to identify the maximum likelihood estimates in an SDM model (a hedonic home price model, with observations being 5,000 individual homes), $$y=\rho W y + X\beta + WX\lambda + \epsilon$$ ...
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