# Questions tagged [determinant]

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

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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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### What does the determinant of a homography matrix represent?

I am trying to find one image (needle) within another (haystack). I am using the following OpenCV method, which first matches keypoints with SIFT and then applies homography: https://opencv-python-...
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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....
1 vote
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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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### 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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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 $\... 4 votes 1 answer 320 views ### Determinant of a block matrix with sparse elements I have a positive definite symmetric matrix that looks like $$\pmatrix{A & 0 & 0 & E \\ 0 & B & 0 & F \\ 0 & 0 & C & G \\ E^\prime & F^\prime & G^\prime &... • 2,164 0 votes 1 answer 319 views ### How to get the determinant of a covariance matrix from its diagonal elements I am trying to implement a speaker recognition system in MATLAB. I am using Gaussian Mixture Models (GMM) for speaker modelling and maximizing the posterior probabilities for classification. The ... • 103 2 votes 0 answers 221 views ### Stability of VAR(p): Prove \det(I_{Kp}-\tilde{A}z)=\det(I_k-A_1z-\ldots-A_pz^p) Given we have a VAR(p) process written in the companion form$$\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 &... • 1,330 7 votes 1 answer 1k views ### 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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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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### 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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### 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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### 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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